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PB-111586-R 


HANDBOOK OF PIEZOELECTRIC CRYSTALS FOR RADIO 
EQUIPMENT DESIGNERS 

John P. Buchanan 

Philco Corporation 

October 1956 


DISTRIBUTED BY: 



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FOR FFKRAL SOCNTIFC AND TECHNICAl INFORMATION 


U. 8. DEPARTMENT OF COMMERCE / NATIONAL BUREAU OF STANDARDS / INSTITUTE FOR APPLIED TECHNOLOGY 



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NOTICE 


This is your copy of the latest edition of "Handbook of Piezo Electric 
Crystals for Radio Equipment Designers" prepared by Phllco for the Air Force 
as a WADC Technical Report. The Report WADC TR-56-156 covers material consid- 
ered useful in the design of crystal oscillators for electronic equipment. 
It will serve well in the hands of the electronic design engineers of your 
organization. The report applies to the application of "Military Type" crystal 
units in equipments designed for the Armed Services. The report strongly in- 
dorses the use of military type crystal units, however, does not discourage 
use of non-military types if a need exists. It does recommend that your organ- 
ization bring to the attention of the "Frequency Control Group" of the indivi- 
dual service organization needs for non-military types before the design 
application is frozen so that military types currently in development may in 
turn be offered for your consideration. This cooperation is strongly requested 
so as to effectively make use of the military types which have established 
sources of supply. In the event this action is not taken, the special, non- 
military, type crystal unit may not be procurable when production quantities 
are most needed. Your support and cooperation is appreciated. 

Sincerely yours 

OwHN B. RIPPERE 
Colonel, USAF 

Chief, Comm & Nav Laboratory 
Directorate of Development 
Wright Air Development Center 






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WADC TECHNICAL REPORT 56-156 
AST I A DOCUMENT NO. AD 110448 


HANDBOOK 

OF 

PIEZOELECTRIC CRYSTALS 

FOR 

RADIO EQUIPMENT DESIGNERS 

This Report Supersedes WADC TR 54-248, Dated December 1954 


Jokn P. Buckanan 
Pkilco Corporation 


OCTOBER 1956 


Communications and Navigation Laboratory 
Contract No. AF 33(616)-2453 
ARDC PROJECT 4155, TASK No. 43033 


W^ri^kt Air Development Center 
Air Researck an«l Development Command 
United States Air Force 
Wri^kt-Patterson Air Force Base, Okio 


McGregor C Werner Midwest Corp, , Otytoa, O. 
2>37-1000-10-13-Sa 



FOREWORD 


This handbook was prepared by the Technical Publications Department, 
TechRep Division, Philco Corporation under Contract AF33 (616) -2453. 
Mr. F. W. Wojcicki served as project director, with Mr. M. W. Nachman 
assuming these duties during the final processing stage. Mr. J. P. 
Buchanan was project engineer and author. Credit for assembling the 
data on specific crystal units and holders in Sections II and III belongs to 
Mr. C. W. Henry. 

This Task No. 43033, titled Handbook of Piezoelectric Crystals for Radio 
Equipment Designers, ARDC Project 4155 was administered under Mr. 
V. J. Carpantier as chief of the Specialties Section, Communication 
Branch of the Communications and Navigation Laboratory, Wright Air 
Development Center. 

Appreciation is extended to Gentile Air Force Depot, Squier Sigrnal Lab- 
oratory, Armour Research Foundation at Illinois Institute of Technology, 
New York University College of Engineering, and to the many individuals 
and other organizations whose generous cooperation has proved so im- 
portant during the preparation of the handbook. In particular, the suc- 
cessful conclusion of the project is heavily indebted to the interest, admin- 
istrative assistance, and many valuable suggestions of Mr. E. H. Borgelt 
of the Frequency Control Group, Wright Air Development Center, and to 
Mr. R. A. Sykes and assistants at Bell Telephone Laboratories for freely 
giving of their time and knowledge in reviewing the text and contributing 
important corrective comments for improving the usefulness and accuracy 
of the text. 


COPYRIGHT CREDITS 

Many of the illustrations used in the general information 
section of the handbook are copyrighted by D. Van Nos- . 
trand Co., 250 Fourth Ave., New York, N. Y. and are iden- 
tified by an asterisk (*) after each title. Reproduction of 
any of these photographs is prohibited without the express 
permission of the copyright holder. 


COPIES 

Copies of this report may be obtained from the Office of Tech- 
nical Services, U. S. Department of Commerce, Washington, D. C. 
at a nominal cost. 


WADC TR 56-156 



ABSTRACT 



A comprehensive manual of piezoelectric control of radio frequencies is 
offered. It is directed toward the design of oscillator circuits having opti- 
mum operating conditions when employing Military Standard crystal units. 
Included is a survey of the development of the piezoelectric crystal art; 
descriptions and characteristics of all crystal elements and mounting 
methods that have found commercial application ; a detailed study of the 
equivalent circuit characteristics of crystal units; analyses of basic piezo- 
electric oscillator principles and of the effects of changes in various circuit 
parameters, using the Pierce oscillator as a reference circuit ; analyses and 
recommended design procedures for all types of piezoelectric oscillator cir- 
cuits used, or tested for use, in USAF equipments; schematic diagrams 
and tables giving actual circuit parameters of all available nonclassified 
piezoelectric oscillators now being used in USAF equipments ; descriptions 
of all crystal units and crystal holders now being used in USAF equip- 
ments, containing references and schematics of circuits employing those 
crystal units recommended for equipments of new design; a brief discus- 
sion of crystal ovens and descriptions of ovens currently available for use 
with Military Standard crystal units ; and a comprehensive index to increase 
the utility of the handbook as a reference manual. Circuit analyses, deriva- 
tions of equations, and suggestions for design innovations whose sources 
are not directly acknowledged have originated with the author and so far 
as is known have not been specifically confirmed in practice. 


PUBLICATION REVIEW 
This report has been reviewed and is approved. 

FOR THE COMMANDER; 



Col onel , 


c^. 

RIPPERE 

USAF 


Chief, Comm & Nav Laboratory 
Directorate of Development 
Wright Air Development Center 


WADC TR 56-156 


iii 


TABLE OF CONTENTS 


SECTION I— GENERAL INFORMATION 1 

SECTION II— CRYSTAL UNITS 430 

SECTION III— CRYSTAL HOLDERS 569 

SECTION IV— CRYSTAL OVENS 601 

APPENDIX I— ACKNOWLEDGMENTS 629 

APPENDIX II— BIBLIOGRAPHY. ....... 635 

APPENDIX III— LIST OF MANUFACTURERS ... 654 

APPENDIX IV— RELATED SPECIFICATIONS, STANDARDS, 

PUBLICATIONS AND DRAWINGS. 658 

APPENDIX V— DEFINITIONS OF ABBREVIATIONS 

AND SYMBOLS 662 

INDEX 683 


WADC TR 56-156 


iv 



SECTION I— GENERAL INFORMATION 

TABLE OF CONTENTS 

Subject Page 

INTRODUCTION 

Purpose and Scope of Manual 1 

Control of Radio Frequency 1 

The Piezoelectric Effect 1 

Development of Piezoelectric Devices 2 

PHYSICAL CHARACTERISTICS OF PIEZOELECTRIC 
CRYSTALS 

Descriptions of Useful Piezoelectric Crystals 5 

Theory of Piezoelectricity 12 

Modes of Vibration 15 

Orientation of Crystal Cuts 18 

Piezoelectric Elements 20 

STANDARD QUARTZ ELEMENTS 

Types of Cuts 21 

The X Group 22 

The Y Group 34 

FABRICATION OF CRYSTAL UNITS 

Initial Inspection of Raw Quartz 45 

Inspection for Optic Axis and Optical Twinning 46 

Use of Conoscope for Exact Determination of Optic Axis 46 

Sectioning the Stone 47 

Determination of X Axis 49 

Cutting X Block 50 

Determination of Twinning 51 

Preparation of Wafers 51 

Preparation of Crystal Blanks 51 

Methods of Mounting Crystal Blanks in Crystal Holders 52 

Housing of Crystal Units 67 

Aging of Crystal Units 67 

ELECTRICAL PARAMETERS OF CRYSTAL UNITS 

Equivalent Circuit of Crystal Unit 70 

Simplified Equivalent Circuit of Air-Gap Crystal Unit 71 

Effect of R-F Leakage Resistance 71 

Effect of Distributed Inductance 72 

Effect of Distributed Capacitance 73 

Effect of Distributed Resistance 74 


WADC TR 56-156 



TABLE OF CONTENTS— Confinued 


Subject Page 

ELECTRICAL PARAMETERS OF CRYSTAL UNITS (Cont) 

Rule-of-Thumb Equations for Estimating Parameters 74 

Impedance Cliaracteristics versus Frequency 77 

Resonant Frequency of Crystal Unit 78 

Antiresonant Frequency of Crystal Unit 80 

Impedance Curves of Crystal Unit 80 

Parallel-Resonant Frequency, f,,, of Crystal Unit 81 

Typical Operating Characteristics of Crystal Unit 85 

Measurement of Crystal Parameters 86 

Methods for Expressing the Relative Performance of Crystal Units 90 

Activity Quality of Crystal Unit 91 

Frequency Stabilization Quality of Crystal Unit 98 

Bandwidth and Selectivity Parameter of Crystal Unit 102 

Crystal Quality Factor, Q 103 

Stability of Crystal Parameters 105 

CRYSTAL OSCILLATORS 

Fundamental Principles of Oscillators 113 

Fundamental Requirements of Stable Forced-Free Oscillations 113 

Application of Fundamental Oscillator Principles in the Design of 

Electronic Oscillators 114 

Phase Rotation in Vacuum-Tube Oscillators 116 

Types of Crystal Oscillators 119 

Parallel-Resonant Crystal Oscillators 119 

The Pierce Oscillator 120 

The Miller Oscillator 187 

Two-Tube Parallel-Resonant Crystal Oscillator 218 

Oscillators with Crystals Having Two Sets of Electrodes 222 

Crystal and Magic-Eye Resonance Indicator 223 

Series-Resonant Crystal Oscillators 225 

Meacham Bridge Oscillator 226 

Capacitance-Bridge Oscillators 236 

The Butler Oscillator 244 

Transformer-Coupled Oscillator 258 

Grounded-Grid Oscillator 266 

The Grounded-Plate Oscillator 274 

Transitron Crystal Oscillator 276 

Impedance-Inverting Crystal Oscillators 277 

Grounded-Cathode Two-Stage Feedback Oscillator 280 

Colpitts Oscillators Modified for Crystal Control 281 

Crystal Calibration 286 

Synthesizing Circuits 288 

Transistor Oscillators 342 

Packaged Crystal Oscillators 356 

Factors Involved in Oscillator Limiting 370 • 

CROSS INDEX OF CRYSTAL-OSCILLATOR SUBJECTS 375 


WADC TR 56-156 



Section I 
Introduction 


SECTION I— GENERAL INFORMATION 

INTRODUCTION 


PURPOSE AND SCOPE OF MANUAL 

1-1. The purpose of this manual is to provide the 
design and developmental engineer of military 
electronic equipment with a reference handbook 
containing background material, circuit theory, 
and components data related to the application of 
piezoelectric crystals for the control of radio 
frequencies. 

1-2. This manual is composed of the following 
sections : 

I. GENERAL INFORMATION 

II. CRYSTAL UNITS 

III. CRYSTAL HOI.DERS 

IV. CRYSTAL OVENS 

V. APPENDIXES 

1-3. Section I contains a brief historical account 
of the discovery of the piezoelectric effect and of 
the application of crystal resonators as frequency- 
control devices, discussions covering the theory 
and physical properties of piezoelectric crystals, 
descriptions and performance characteristics of 
the more important quartz crystal elements, gen- 
eral discussions of the various crystal-unit fabri- 
cation processes and types of mounting, detailed 
discussions of the equivalent electrical parameters 
and performance characteristics of crystal units, 
and comprehensive qualitative and mathematical 
analyses of the various types of crystal oscillators, 
summarized with recommended design procedures. 
1-4. Sections II, III, and IV provide the technical 
and logistical data, and information concerning the 
application of the crystal units, crystal holders, 
and crystal ovens currently recommended for use 
in equipments of new design. 

1-5. The Appendixes contain the acknowledg- 
ments; a bibliography; a list of manufacturers 
associated with the piezoelectric crystal industry; 
a list of related U. S. Government specifications, 
standards, and publications ; a table of definitions 
for the abbreviations and symbols used in the 
Handbook ; conversion charts ; and an alphabetical 
index. 

CONTROL OF RADIO FREQUENCY 

1-6. The greatly increased demand for military 
radio channels, with the consequent crowding of 


the radio-frequency spectrum, is, in the final 
analysis, a problem for the design engineer of fre- 
quency-control circuits. The problem is essentially 
one of providing a maximum frequency stability 
of the carrier at the transmitting station, and a 
maximum rejection of all but the desired channel 
at the receiving station. In each instance optimum 
results are obtained by the use of electromechani- 
cal resonators — maximum carrier stability is 
achieved by the use of crystal master oscillators, 
and maximum receiver selectivity is achieved by 
the use of crystal heterodyne oscillators and crys- 
tal band-pass filters. 

1-7. The design of a constant-frequency generator 
has been an ideal of radio engineers almost from 
the beginning of radio science. Although many 
purely electrical oscillators have been devised 
which closely approach the ideal, none surpass the 
performance of the high-quality circuits employ- 
ing mechanical oscillators. Temperature-controlled 
oscillators having a sonic-frequency tuning fork as 
the frequency controlling element and followed by 
a number of frequency multiplying stages were 
the first of the radio-frequency generators employ- 
ing the high precision of mechanical control. The 
cumbersomeness and expense of the many multi- 
plier stages, however, have made the tuning fork 
oscillators impracticable insofar as the control of 
any but sonic frequencies are concerned. Today, 
precision control of radio frequencies has been 
made possible through the development of piezo- 
electric resonators, where the frequency-control- 
ling elements, usually quartz plates, have normal 
vibrations in the radio-frequency range. 

THE FIEZOELECTRIC EFFECT 

1-8. The word piezoelectricity (the first two syl- 
lables are pronounced pie-ee') means “pressure- 
electricity,” the prefix piezo- being derived from 
the Greek word piezein, meaning “to press.” 

1-9. “Piezoelectricity” was first suggested in 1881 
by Hankel as a name for the phenomenon by which 
certain crystals exhibit electrical polarity when 
subjected to mechanical pressure. 

1-10. That such a phenomenon probably existed 
seems to have been suggested first by Coulomb in 




WADC TR 54-248 


1 


QUARTZ CHARACTERISTICS 


S«clien I 
Introduction 


the latter part of the 18th century. His suggestions 
prompted Haiiy, and later A. C. Bequerel, into 
undertaking a series of experiments to see if elec- 
tric effects could be produced purely by mechanical 
pressure. Although both Haiiy and Bequerel re- 
ported positive results, there is some doubt as to 
whether these were not due to contact potentials 
rather than to piezoelectric properties of the sub- 
stances investigated — particularly since electrical 
polarities were reported in crystals that are now 
known to be non-piezoelectric. 

1-11. It is to the Curie brothers, Jacques and 
Pierre, that the honor goes for having been the 
first (in 1880) to verify the existence of the piezo- 
electric effect. (For the initial report of their 
discovery, see paragraph 1-56.) 

1-12. The Curie brothers tested a number of crys- 
tals by cutting them into small plates that were 
then fitted with tin-foil electrodes for connection 
to an electrometer. When subjected to mechanical 
pressure, several of the crystals caused the leaves 
of the electrometer to be deflected. Among those 
crystals showing electrical polarities were quartz, 
tourmaline, Rochelle salt, and cane sugar. In the 
year following these experiments, a prediction by 
Lippmann that the effect would prove reversible 
prompted the Curies to further investigations. The 
results verified Lippmann’s prediction by revealing 
that the application of electric potentials across a 
piezoelectric crystal would cause deformations in 
the crystal which would change in sign with a 
change in electric polarity. Furthermore, it was 
found that the piezoelectric constant of propor- 
tionality between the electrical and mechanical 
variables was the same for both the direct (pres- 
sure-to-electric) and the converse effects. In other 
words, the same polarization at the surface of the 
electrodes that results from a particular deforma- 
tion of the crystal can, in turn, if applied from an 
external source, produce the deformation. 

1-13. It should be mentioned that the piezoelectric 
effect, which occurs only in certain asymmetrical 
crystals, is not to be confused with electrostric- 
ture, a property common to all dielectrics. Al- 
though electrostricture is a deformation of a di- 
electric produced by electric stress, it is unlike the 
converse piezoelectric effect in that its magnitude 
varies, not linearly with the electric field, but with 
the square of the field, and is unaffected by a 
change in the applied polarity. Electrostricture is 
the type of deformation a capacitor undergoes 
on being charged. In piezoelectric crystals this 
effect is normally small compared with the piezo- 
electric pi operties. 


DEVELOPMENT OP PIEZOELECTRIC DEVICES 

1-14. From the time of its discovery until World 
War I, the piezoelectric effect found few practical 
uses. Those applications it did find appeared in the 
form of occasional laboratory devices for measur- 
ing pressure or electric charges. For the most part, 
however, little attention was attracted to piezo- 
electricity outside the crystallographer’s study. 
Nevertheless, during this time considerable theo- 
retical progress was made, due chiefly to the 
efforts of Lord Kelvin, Duhem, Pockels, and 
Woldmar Voigt. Voigt’s comprehensive Lehrhuch 
der Kristallphysik, published in 1910, is still con- 
sidered the reference authority on the mathe- 
matical relationships among crystal variables. 
1-15. It was after the outbreak of World War I 
before serious attention was given to the practical 
application of piezoelectric crystals. During the 
war Professor Paul Langevin of France initiated 
experiments with the use of quartz crystal plates 
as underwater detectors and transmitters of 
acoustic waves. Although Langevin’s immediate 
purpose was to develop a submarine detecting de- 
vice, his research became of vital importance to 
many other developments. Not only did it attract 
the applied sciences to the possibilities of piezo- 
electric crystals, but also it initiated the modem 
science of ultrasonics. 

1-16. The detecting apparatus that Langevin 
eventually devised employed quartz “sandwiches” 
which were coupled electrically to vacuum-tube 
circuits, and could be exposed under water where 
they would vibrate at the frequency of an applied 
voltage, or at the frequency of an incident acous- 
tic wave. The first function was employed to emit 
ultrasonic waves, and the second function to re- 
ceive and reconvert the echo into electrical energy 
for detection. 

1-17. At the same time that Langevin was experi- 
menting with quartz as a supersonic emitter and 
detector, Dr. A. M. Nicolson, at Bell Telephone 
Laboratories, was independently investigating the 
use of Rochelle salt to perform the same functions 
at sonic frequencies. Indeed, his first application 
for a patent on a number of piezoelectric acoustic 
devices, April 1918, preceded by five months Lan- 
gevin’s initial application for a French patent. 
Employing Rochelle salt instead of quartz, because 
of its greater piezoelectric sensitivity, Nicolson 
constructed a number of microphones, loud- 
speakers, phonograph pickups, and the like. 
Among the circuits included in his 1918 patent 
application, was one that later proved of particu- 
lar interest — an oscillator employing a Rochelle 


WADC TR 56-156 


2 



Section I 
Introduction 


salt crystal as shown in figure 1-1. With this ex- 
ception, all the early applications of the piezoelec- 
tric crystal involved its use as a simple electro- 
mechanical transducer. That is, it was used either 
to transform mechanical energy in one system to 
electrical energy in another, or vice versa. Nicol- 
son’s oscillator was a distinct innovation in that it 
employed a piezoelectric crystal as a transformer 
of electrical energy to mechanical energy and back 
to electrical energy. 

1-18. When Nicolson devised his oscillator, none 
of the possible functions of a piezoelectric vibrator 
had previously been investigated or discussed. His 
patent application offered no description of the 
crystal’s function, although presumably the crys- 
tal performed in some way to transfer part of the 
plate circuit energy to the grid circuit. Evidence 
that the normal vibrations of the crystal actually 
controlled the frequency seems to have existed, but 
no mention was made of this fact. The circuit, 
however, embodies the combined principles of 
coupler, filter, and resonator. Obviously the crystal 
acts as a coupler between the plate and grid cir- 
cuits ; and, inasmuch as the crystal may block the 
feedback of all plate energy except that at the 
frequency of the crystal’s normal mode of vibra- 
tion, the crystal may be imagined to perform the 
function of a filter, even though the over-all opera- 
tion is that of an oscillator. Finally, if the plate 


tap is connected at the bottom of the coil, so that 
the only feedback is through the plate-to-grid 
capacitance of the vacuum tube, the crystal may 
function as a conventional resonator, controlling 
the frequency as would a tuned grid tank circuit — 
the complete vacuum-tube circuit being the equiv- 
alent of a tuned-plate, tuned-grid oscillator. Thus, 
to Dr. Nicolson belongs the honor of being the 
first to employ the piezoelectric crystal purely as a 
circuit element, in all its principal circuit functions. 
1-19. Although Nicolson was the father of the 
piezoelectric crystal circuit. Professor Walter G. 
Cady, of Wesleyan University, was its greatest 
prophet. In 1918 during a series of experiments 
being conducted to investigate the use of Rochelle 
salt plates for underwater signaling. Dr. Cady be- 
came interested in the electromechanical behavior 
of crystals vibrating in their normal modes. Out 
of the resonant properties that he discovered, he 
came to visualize the great possibilities that the 
piezoelectric crystal afforded as a resonator of 
high stability. After experimenting with several 
circuits, including the first quartz-controlled oscil- 
lator, Dr. Cady, in January 1920, not aware that 
Dr. Nicolson considered his oscillator controlled by 
the resonance of its crystal, submitted a patent 
application for the piezoelectric resonator, in which 
he reported its possibilities as a frequency stand- 
ard, filter, and coupler, and described the principles 



figura 1-1. The first crystal- (Rochelle salt) controlled oscillator. Invented by A. M. Nicolson, 1918 


WADC TR 56-156 


3 



Section I 
Introduction 


of its operation. Although subsequent litigation 
concerning the discovery of the piezoelectric-reso- 
nator principle was decided in Dr. Nicolson’s favor, 
it is distinctly to Dr. Cady’s credit that he was the 
first to fully grasp the import of the piezoelectric 
resonator and to publish a public report of its prin- 
ciples and possibilities. His early pioneering in the 
field and his many later contributions have made 
Dr. Cady the American dean of piezoelectricity. 
1-20. It soon became apparent that quartz crystals 
were the most stable and practical for use as reso- 
nators. Many investigators were attracted to the 
field, and progress was made both in the design 
and theory of crystal circuits. Professor G. W. 
Pierce of Harvard showed that quartz crystal os- 
cillators could be constructed with a single ampli- 
fier stage, as Nicolson had already done using 
Rochelle salt. This marked a considerable improve- 
ment over Cady’s oscillators, which had consisted 
of two or more vacuum-tube stages. Of particular 
note was the analysis by K. S. Van Dyke, in 1925, 
of the electrodynamic characteristics of a crystal 
resonator in terms of an equivalent electrical net- 
work : for the first time a way was opened to an 
understanding of the crystal resonator. In 1928, 
E. M. Terry showed that the frequency of a crystal 
oscillator was not entirely controlled by the crys- 
tal characteristics, but to a small degree was also 
dependent upon the other circuit constants. F. B. 
Llewellyn, in 1931, presented a classic analysis of 
oscillators showing the circuit impedance relation- 
ships that are necessary if the frequency is to be 
independent of variations in the voltage supply 
and vacuum-tube characteristics. Although the 
subject matter of this treatise deals with oscilla- 
tors in general, the principles are applicable to the 
design of crystal oscillators, if the electrical pa- 
rameters of the crystal are known. 

1-21. The tuned-circuit oscillators of the early 
transmitters normally operated with heavy and 
variable loads. Many of the oscillators operated 
directly into an antenna, and in broadcast trans- 
mitters, modulation was performed in the oscilla- 
tor stage. This resulted in considerable frequency 
instability, and broadcast reception was often un- 
intelligible because of the frequency difference in 
radio waves arriving by different paths. It was in 
the determination of the cause and the correction 
of such interference that Messrs. R. Bown, D. K. 
Martin, and R. K. Potter of the Research and De- 
velopment Department of the American Telephone 
and Telegraph Company recommended the use of 
lightly loaded crystal-controlled oscillators fol- 
lowed by amplifiers. Under their supervision. Sta- 
tion WEAF in New York, in 1926, became the first 


crystal-controlled broadcasting station. 

1-22. The principal factor limiting the stability of 
the early quartz oscillator was the relatively large 
frequency-temperature coefficient of the crystal, 
which allowed small changes in the ambient tem- 
perature to cause excessive changes in the reso- 
nant frequency. The immediate method of obtain- 
ing stability, of course, was to mount the crystal 
in an oven where the temperature could be con- 
trolled thermostatically. However, to decrease the 
temperature coefficient of the crystal, itself, also 
became the goal of a number of researchers. Be- 
cause some quartz plates exhibited positive tem- 
perature coefficients, whereas others exhibited 
negative coefficients, according to the orientation 
of the plate with respect to the axes of the mother 
crystal, the possibility arose that there should be 
some shape or median angle of cut which would 
have a zero coefficient. The first empiricists to en- 
ter the field were E. Giebe and A. Scheibe in Ger- 
many. In the United States, Mr. W. A. Marrison 
of Bell Telephone Laboratories turned his atten- 
tion to the problem of achieving the maximum 
precision possible in frequency control, and by 
1929 had perfected a 100-kc frequency standard 
using a doughnut-shaped crystal (originally pio- 
neered by Giebe) with a nearly zero temperature 
coefficient. This success encouraged the Bell Lab- 
oratories research staff to launch a Concerted 
investigation into all phases of quartz crystal 
physics. Out of this program have arisen most of 
the principal advances in the design and produc- 
tion of quartz crystal units in the United States; 
although the early pioneering of S. A. Bokovoy and 
C. F. Baldwin at RCA has also been of notable 
significance. 

1-23. Originally only the Curie, or X-cut, quartz 
plate was used — a plate in which the thickness 
dimension is parallel to the crystal’s X axis. Later 
the Y cut, where the thickness dimension is paral- 
lel to a Y axis, developed by E. D. Tillyer of the 
American Optical Co., began to compete with the 
Curie cut as the frequency-control element in com- 
mercial oscillators. By 1934, Messrs. F. R. Lack, 
G. W. Williard, and I. E. Fair of Bell Telephone 
Laboratories announced the discovery and devel- 
opment of two types of plates, called the AT and 
BT cuts, with such small temperature coefficients 
that they could operate stably under normal con- 
ditions without the use of temperature-controlled 
ovens. Concurrently, Bokovoy and Baldwin at RCA 
were experimenting with a series of crystals that 
they named the V cut, and their work, although of 
a less rigorous theoretical approach, substantially 
paralleled much of the research that was done at 


WADC TR 56-156 


4 



Bell Laboratories. In 1937, Messrs. G. W. Williard 
and S. C. Right announced the development of the 
CT, DT, ET, and FT cuts ; and by 1940 Mr. W. P. 
Mason had discovered the GT cut, the most stable 
resonator ever devised. The time-keeping stand- 
ards at both the Greenwich Observatory and the 
U. S. Bureau of Standards now use this crystal. 
Where other cuts exhibit a zero temperature co- 
efficient only at certain temperatures, the GT cat 
has almost a zero temperature coefficient over a 
range of 100°C. Besides the cuts discussed above, 
a number of others have been investigated which 
have proved particularly applicable for special 
uses. Among these are the AC, BC, MT, NT, 5- 
degree X, and the — 18-degree X cuts. 

1-24. Paralleling the development of the new crys- 
tal cuts were the improvements made in the design 
of crystal holders. The early holders provided no 
means of “clamping” a crystal, for they were de- 
signed originally to accommodate X-cut plates 
whose favored modes of vibration required that the 
edges be free to move. Since the crystal in such 
a holder will slide about if used in equipment sub- 
ject to mechanical vibrations, a method of clamp- 
ing was needed before the crystal could be used 
in vehicular or airborne radio sets. Mr. G. M. 
Thurston of Bell Telephone Laboratories was led 
to the solution of this problem when he discovered 
that a crystal would not be restricted if clamped 
only at the mechanical nodes of its normal vibra- 
tions. The exact positions of these points, where 
the standing-wave amplitude is zero, depend, of 
course, on the particular mode of vibration. The 
low-frequency —18-degree X-cut crystal, for in- 
stance, can be held by knife-edged clamps running 
along its center, whereas AT- and BT-cut crystals 
can be clamped at their comers. Cantilever and 
wire supports which resonate at the crystal fre- 
quency have been devised for holding crystals at 
their centers. Although the mounting of crystals 
requires a far more exacting technique than for- 


Sacilon I 

Physical Characteristics of Piezoelectric Crystals 

merly, the crystal holder today provides support 
and protection sufficient to insure high perform- 
ance stability, even under the severe conditions of 
vibration that exist in military aircraft and tanks. 
1-25. Unfortunately, the extremely critical nature 
of the design and production of crystal units has 
made it impracticable for manufacturers to mass- 
produce units with such exactitude that all the 
equivalent electrical parameters are standardized 
with an accuracy comparable to that now achieved 
in the case of vacuum tubes or other circuit com- 
ponents. However, definite progress has been 
made in this direction, and, if the need warrants 
the additional cost, reasonably exact characteris- 
tics may be obtained. For several years, each 
crystal unit had to be tested in a duplicate of the 
actual circuit in which it was to be used. This pro- 
cedure was disadvantageous from the points of 
view of both the radio design engineer and the 
crystal manufacturer. On the one hand, the radio 
engineer, knowing little more than the nominal 
frequency of the crystal unit to be installed in his 
circuit, could not achieve that degree of perfection 
in oscillator design which was otherwise theoret- 
ically possible. On the other hand, the task of 
making a given oscillator perform correctly effec- 
tively became the responsibility of the crystal 
manufacturer, since it was necessary for him to 
fit each crystal unit by trial and error to the par- 
ticular circuit for which it was intended. In recent 
years this cut-and-try procedure has been allevi- 
ated considerably by the development of standard 
test sets and by improvements in production tech- 
niques that permit more critical specifications. It 
is hoped that this handbook, by providing a more 
comprehensive description of the technical char- 
acteristics of the crystal units recommended for 
new design, will contribute in removing the limita- 
tions that too often in the past have forced the 
practical design engineer to approach his crystal 
circuits philosophically, rather than scientifically. 


PHYSICAL CHARACTERISTICS OF PIEZOELECTRIC CRYSTALS 


DESCRIPTIONS OF USEFUL PIEZOELECTRIC 
CRYSTALS 

1-26. The piezoelectric effect is a property of a 
non-conducting solid having a crystal lattice that 
lacks a center of symmetry. Of the 32 classes of 
symmetry in crystals, 20 are theoretically piezo- 
electric, and the actual crystals which have been 
found in this category are numbered in the low 
hundreds. 


1-27. Until the time of World War II only three 
crystals were commercially employed for their 
piezoelectric properties — quartz, Rochelle salt, 
and tourmaline. Today, the number is being in- 
creased by the development and application of 
synthetic crystals. Of these, the principal ones 
used in frequency selective circuits are ethylene 
diamine tartrate (EDT), dipotassium tartrate 
(DKT), and ammonium dihydrogen phosphate 
(ADP). See figure 1-2. 


WADC TR 56-156 


5 


Mction I 

Physical Characteristics of Piezoelectric Crystals 



Figur* 1-2. Commercially used piezoelectric crystals other than quarts 


WADC TR 56-156 


6 



1-28. Piezoelectricity is still in its infancy, and 
until more data have been collected and coordinated 
into a comprehensive atomic theory of the phe- 
nomenon, the chemist will have few clues to direct 
his search for a crystal having the maximum 
possible piezoelectric effect. 

Tourmalin* 

1-29. Tourmaline is a semiprecious stone which at 
one time was called the “Ceylon Magnet.” This title 
seems to have been given it by early 18th century 
traders who introduced the stone to Europe, with 
the story of its strange magnetic property. If 
placed in hot ashes, tourmaline behaves as if it 
were electrified — first attracting ashes and then 
throwing them off. This is the phenomenon of 
pyroelectricity, closely associated with piezoelec- 
tricity, and was possibly the first electrical effect, 
other than lightning and St. Elmo’s fire, ever to be 
noticed by man. According to the theory proposed 
by Lord Kelvin, the pyroelectric effect of tourma- 
line is due to a permanently polarized lattice in the 
crystal, so that when heated, an unneutralized in- 
crease in the dipole moment occurs, proportional 
to the change in temperature and the coefficient of 
expansion. It was this pyroelectric theory of 
permanently polarized crystals that eventually 
prompted the Curie brothers to test for the piezo- 
electric effect. 

1-30. Tourmaline is unsuitable for wide com- 
mercial use because of its expense and the scarcity 
in the number of large-sized natural crystals. Also, 
the temperature coefficients are negative for all 
tested modes of vibrations, which fact rules out 
the possibility of zero-coefficient cuts. 

1-31. Tourmaline does have the advantage of dura- 
bility and a large thickness-frequency coefficient, 
so that for a given frequency it permits a more 
rugged crystal unit than quartz. For this reason 
it is sometimes used for the control of very high 
frequencies. However, the chief piezoelectric ap- 
plication of tourmaline is in the measuring of 
hydrostatic pressures. 

Rochell* Salt 

1-32. Rochelle salt (NaKC 4 H 40 «- 4 HsO) is sodium 
potassium tartrate with four molecules of water of 
crystallization. The crystals are grown commer- 
cially by seeding saturated solutions of the salt 
and decreasing the temperature of the solutions a 
few tenths of a degree per day. They were first syn- 
thesized in 1672 by Pierre Seignette, an apothecary 
of La Rochelle, France, and until the time of Nicol- 
son’s inventions the salt was used primarily for its 


Section I 

Physical Characteristics of Piezoelectric Crystals 

medicinal value. Its exceptionally great piezoelec- 
tric effect — a blow with a hammer can generate as 
much as five thousand volts — has made Rochelle 
salt the principal crystal for use as a transducer in 
acoustic devices, such as microphones, loud- 
speakers, pickups, hearing aids, and the like. As a 
stable resonator it is far inferior to quartz, not 
only because of a greater sensitivity to tempera- 
ture variations, but also because of its tendency to 
disintegrate during extremes of humidity. If the 
ambient humidity drops below 35 per cent at room 
temperature, the water of crystallization will be- 
gin to evaporate, leaving a dehydrated powder on 
the crystal surface. Should the humidity rise above 
85 per cent at room temperature, the salt will 
absorb moisture and begin to dissolve. For these 
reasons a Rochelle salt crystal should be mounted 
in a hermetically sealed container, or, if this is not 
possible, at least coated with wax. In the case of 
the former, if powders of both the crystalline and 
dehydrated forms of Rochelle salt are also enclosed 
within the sealed chamber, the humidity of the 
chamber will automatically increase or decrease 
with corresponding changes of temperature, and a 
stable balance between the crystal and chamber 
vapor pressures will be maintained. However, at a 
temperature of 55°C (130°F) the crystal, which 
is a double salt of tartaric acid, breaks down into 
sodium tartrate, potassium tartrate, and water. 
The solution formed will remain a viscous liquid 
for some time if super-cooled, and, as such, makes 
an effective glue for binding together plates of the 
crystal. 

1-33. Although Rochelle salt, between the tem- 
peratures of --18°C and -|-24“C, has a greater 
piezoelectric effect than any other crystal, it seems 
that eventually it will be replaced by other syn- 
thetic crystals, in particular, ADP (NH 4 H 2 PO 4 ), 
which requires no water of crystallization. Never- 
theless, as an electromechanical transducer, Ro- 
chelle salt is still the most widely used of the 
piezoelectric crystals. 

ADP 

1-34. ADP (NHiHjPO,), ammonium dihydrogen 
phosphate, was discovered and used during World 
War II as a substitute for Rochelle salt in under- 
water sound transducers. Like Rochelle salt, ADP 
crystals can be grown commercially; but unlike 
Rochelle salt, it requires no water of crystalliza- 
tion, and hence has no dehydration limitations, 
being able to stand temperatures up to 100°C 
(212°F). Also, ADP is more durable mechanically 
than Rochelle salt. 

1-35. Although the crystal’s principal application 


WADC TR 56-156 


7 



Section I 

Physical Characteristics of Piezoelectric Crystals 

has been in submarine-detecting apparatus, its 
greater stability suggests the probability that it 
will eventually replace Rochelle salt as the prin- 
cipal transducer in other sonic devices. 

EDT 

1-36. EDT (CeHijNzOs), ethylene diamine tar- 
trate, was discovered and developed during World 
War 11 as a substitute for quartz in low-frequency 
filter units. Quartz crystals at this time were in 
such great demand for the frequency control of 
military communication equipment, that a short- 
age developed in the supply of large-sized natural 
crystals which were needed for cutting filter plates 
of IV 2 to 2 inches in length. This shortage was 
acutely felt in the telephone industry, where there 
exists the chief demand for such plates for use in 
the band-pass filters of carrier systems. The dis- 
covery of EDT was the solution to this problem, 
for this crystal can be grown to any size desired, 
and it has the chemical stability (no water of 
crystallization), low mechanical loss, zero tem- 
perature coefficient, and small aging effects that 
make it a suitable substitute for quartz. 

1-37. EDT is not as rugged mechanically, nor does 
it have quite as high a Q as quartz — although the 
EDT crystal units operating as filter elements in 
the 20- to 180-kc range do have Q’s in the neighbor- 
hood of 30,000. Moreover, for use as the frequency- 
control element in high-frequency oscillators, EDT 
is inferior to quartz because of its greater sensi- 
tivity to temperature changes. Even though 
high-frequency modes of vibration have been 
found with zero temperature coefficients, the 
temperature shift to either side of the optimum 
value must be kept approximately one-fifth that 
for a comparable quartz plate (BT cut, for 
example) in order to maintain the same frequency 
tolerance. Where only a minimum of temperature 
control might be needed for quartz, EDT will 
require fairly accurate control. Because of these 
disadvantages, EDT does not threaten at this time 
to replace quartz in high-frequency oscillators, but 
it does have promising possibilities for use in 
oscillators of the frequency-modulated type. Here, 
EDT plates have the advantage of a relatively wide 
gap between their resonant and antiresonant 
frequencies, thus permitting a large percentage 
swing of the oscillator frequency. If temperature- 
controlled, the EDT crystal can thus give crystal 
stability to a frequency -modulated transmitter. 

DKT 

1-38. DKT (K.CiHjOo-'^HsO), dipotassium tar- 
trate, is another synthetic crystal which was in- 


vestigated at Bell Telephone Laboratories during 
World War II. The DKT molecule is similar chemi- 
cally to that of Rochelle salt except that the so- 
dium atom has been replaced by another potassium 
atom. The crystal, however, differs from Rochelle 
salt in that it contains only one molecule of water 
for each two DKT molecules, as compared with a 
water-to-salt molecular ratio of four-to-one in the 
Rochelle salt crystal, and it exhibits no tendency 
to dehydrate below 80°C (176°F). Also, the piezo- 
electric characteristics of DKT are less like those 
of Rochelle salt than of quartz. Indeed, in the 
lower-frequency filter circuits, DKT seems as 
promising as EDT as a substitute for quartz. 

1-39. As compared with EDT, DKT has the advan- 
tage of better temperature-frequency character- 
istics. Zero temperature coefficients are possible 
where the frequency deviation on either side of the 
zero point is only one-third that for EDT. How- 
ever, DKT crystals are more difficult to grow than 
the EDT crystals, and primarily for this reason the 
development of a small EDT industry has already 
been established, whereas the DKT crystals are 
still in the laboratory stage. 

Raw Quarti 

1-40. Quartz is silicon dioxide (SiOj) crystallized 
in hard, glass-like, six-sided prisms. The normal 
crystal structure is called alpha quartz; if the 
temperature is raised above 573°C (1063°F) most 
of the piezoelectric property is lost with a crystal 
transformation to beta quartz. At 1750°C 
(3182°F) the crystal structure is permanently 
lost, and the melted quartz assumes the fused 
amorphous form of silica. The density of alpha 
quartz at 20°C (68'’F) is 2.649 grams per cubic 
centimeter. The hardness of quartz is rated at 7 
on Mohs’ scale — a greater hardness than glass or 
soft steel, but less than hard steel. 

1-41. Silicon dioxide is believed to constitute ap- 
proximately one-tenth of the earth’s crust. It 
occurs in many crystalline forms such as quartz, 
flint, chalcedony, agate, onyx, etc., and in the fused 
amorphous state of silica, called “quartz glass.” 
Although quartz is an abundant mineral — sand 
and sandstone consist largely of quartz granules — 
large crystals of good quality are to be found in 
only a few areas. The chief source of supply has 
been Brazil, although large deposits of lower 
quality are also to be found in Madagascar and in 
the United States. Progress has been made in 
growing quartz crystals artificially. Such crystals 
are now commercially available, although this 
quartz source is still primarily in the develop- 
mental stage. The tremendous pressures required 


WADC TR 56-156 


8 



and the slow rate of growth have, until very re- 
cently, prevented quartz manufacture from being 
commercially feasible. Advances are now being 
made in growing imperfection-free quartz stones 
having major dimensions so oriented relative to 
the principal crystal axes that a desired type of 
quartz cut can be obtained with minimum waste. 
The future possibilities of quartz manufacture 
appear quite promising. 

1-42. The large quartz crystals of geological origin 
are the products of long ages of growth under 
great pressure. The growing crystal assumes the 
shape of a hexagonal prism with each end pyramid- 
ing to a point. The prismatic faces are designated 
as m faces, see figure 1-4, and adjacent m faces 
always intersect at angles of 120 degrees. The 
opposite m faces of the prism are always parallel, 
but are rarely of the same dimensions. These faces 
are not perfectly planar, but are streaked with 
small horizontal growth lines, or striae. Parallel to 
the growth lines are the bases of the six end faces 
— three r and three z faces — which form a hexag- 
onal pyramid, but with only the r faces meeting at 
the apex. The end faces are quite smooth, with the 
r, or major, faces usually appearing more polished 
than the z, or minor, faces. Figure 1-3 shows a 
mother crystal with one of the pyramidal ends 
missing. Complete crystals are rarely found except 
in very small sizes. More likely both pyramidal 
ends will be missing, and frequently crystals are 



Figure 1-3. Raw <fuartx stone 
WADC TR 56-156 


Section I 

Physical Characteristics of Piezoelectric Crystals 

found with all the natural faces broken or eroded 
away. The largest quartz crystal that has been re- 
corded was found in Brazil. It is described as a 
crystal of smoky quartz, 7 ft 2 in. long, 11 ft 2 in. 
in circumference, and weighing more than 5 tons. 
1-43. Quartz is enantiomorphous — that is, it oc- 
curs in both right-handed and left-handed forms, 
which are mirror images of each other. The 
enantiomorphic faces of two ideal alpha-quartz 
crystals are represented in figure 1-4. The left- 
handed and right-handed forms are indicated by 
the direction in which the small upper x and s 
faces appear to be pointing. Note that this rule is 
valid regardless of which end of the crystal is 
turned up. However, the x and s faces are rarely 
found, so that the handedness of a crystal is 
usually determined by noting the optical effects 
when polarized light is passed through the crystal 
parallel to the optic (lengthwise) axis. 

IMPERFECTIONS IN QUARTZ 

1-44. Pure quartz of structural perfection is a 
transparent, colorless crystal — such that the early 
Greek physicists believed it to be a perfected form 
of ice. Through the centuries quartz has been cut 
and ground into many ornaments, and was mysti- 
cally respected in the ancient art of crystal gazing. 
1-45. The presence of impurities can convert 
quartz into a variety of gem-like colors. Amethyst, 
agate, and jasper are all quartz crystals colored by 
impurities. A different form of coloring is that 
which gives a smoky appearance to quartz. This 
effect differs in degree from crystal to crystal, and 
in extreme cases a crystal may be so dark that it 
cannot be inspected for defects nor for the align- 
ment of axes. However, by heating a smoky crystal 
from 350°C (662°F) to 500°C (932°F) it becomes 

t z 




Figure 1-4. Left and right quartz crystals 


9 


Section I 

Physical Characteristics of Piezoelectric Crystals 

quite as clear as the purest stone. Possibly the 
coloration is due to the dissociation of some of the 
silicon dioxide molecules, which recombine on heat- 
ing ; in any event, crystals which have been cleared 
of smokiness, remain clear, and have the same 
physical properties as the normal colorless crystals. 
1-46. Other than those arising from chemical 
impurities, there are three types of structural 
defects to be avoided when cutting blanks from 
the raw quartz. These are cracks, inclusions, and 
twinning. 

Cracks 

1-47. All raw crystals contain cracks to some 
extent, particularly near their surfaces, where 
fractures are easily caused by impacts. Tempera- 
ture variations and growth conditions are also 
causes of cracking. The larger cracks are readily 
visible, but not the separations with dimensions 
comparable to a wave-length of light. For this 
reason, any detected crevice should be assumed to 
extend somewhat beyond its visible length. Raw 
quartz should be handled with particular care, for 
the large crystals are more vulnerable to fractur- 
ing than are the small finished plates. No finished 
plate, however, should be permitted to contain a 
crack. 

Inclusions 

1-48. Inclusions are small pockets, often sub- 
microscopic, holding foreign matter which was 
entrapped during the crystal’s period of growth. 
The trapped material may be a gas, liquid, solid, 
or any combination thereof. The pockets are often 
too small to be seen individually, but are readily 
detected by the shapes and coloring of the clusters 


they form. Groups of the smallest-size inclusions 
have a bluish cast; groups of medium-size in- 
clusions appear as a white frosting; and the larger 
inclusions are individually visible as small bubbles. 
Some of the clusters appear as small clouds; others 
appear as needles, which may be fine or feathery, 
and which may form parallel rows or spread 
comet-like from a bubble origin ; still other groups 
are draped in sheets or folds like veils; and, finally, 
there are those inclusions that arrange themselves 
in surfaces parallel to the natural crystal faces, 
outlining former growths, and appearing as crystal 
phantoms within a crystal. See figure 1-5. Not a 
great deal is known concerning the effect of in- 
clusions upon the performance of finished plates. 
However, the fine textured (blue) inclusions are 
the least objectionable, and the isolated bubbles are 
more to be tolerated than a veil or phantom. Blue 
needles are permissible in large, low-frequency 
plates that are not to be driven at high levels. 
Nevertheless, any inclusion weakens a crystal, and 
will not be present in a high-quality, finished plate. 

Twinning 

1-49. Twinning is the intergrowth of two crystal 
regions having oppositely oriented axes. This ab- 
normality is rarely detectable by a casual visual 
inspection, and a crystal that appears homo- 
geneous throughout may, indeed, have several 
twinned areas; in fact, almost all large crystals 
have twinning to some extent. There are two types 
of twinning common to quartz — electrical twin- 
ing and optical twinning. In electrical twinning, 
only the electrical sense of the crystal axes is 
reversed, whereas in optical twinning, not only the 
electrical sense, but the handedness of the crystal 



Figure 1-5. Quartz crystal containing inclusions and fractures * 


WADC TR 56-156 


10 



structure is reversed — that is, one area will be 
right-handed and the other left-handed. 

1-50. A finished plate, if it is to have predictable 
characteristics, must be cut entirely from a region 
having the same crystal structure; otherwise, the 
piezoelectric properties of one region will interfere 
with those of the other. Electrical twins are 
usually large, so both areas may be used separately 
for crystal blanks. Optical twinning, on the other 
hand, is usually confined to pockets, which are 
normally too small to provide crystal blanks, them- 
selves, so that only the predominant crystal region 
can be utilized. 

THE AXES OF QUARTZ 

1-51. There are several crystallographic conven- 
tions by which the reference axes of crystals may 
be chosen, and much confusion has resulted in the 
past because of the various preferences of different 
crystallographers. Insofar as the over-all piezo- 
electric properties are concerned, the orientations 
of quartz have been universally measured accord- 
ing to rectangular sets of X, Y, and Z axes, with 
the XY, XZ, and YZ planes determined according 
to the crystal symmetries. However, even in this 
case, the choice of positive and negative axial and 
angular directions for right and left quartz re- 
mained more or less a matter of preference until 
the system proposed by the I.R.E. in 1949 became 
generally adopted. It is the I.R.E. system that will 
be followed here. It should be remarked first, how- 
ever, that a crystal axis is not intended necessarily 
to coincide with a central point in the crystal, but 
may represent any straight line parallel to the 
axial direction. It might also be noted that the 
different types of crystal faces are designated in 
this manual by the small letters m, r, s, x, and z, 
and these should not be confused with the capital 
letters X, Y, Z which denote the axes, nor with the 
small letters, x, y, z, when used to denote dimen- 
sions of a crystal in the axial directions. 

Z Axis 

1-52. The Z axis is the lengthwise direction of the 
quartz prism and is perpendicular to the growth 
lines of all the m faces. It is an axis of three-fold 
symmetry, so that there are three sets of XY axes 
for each crystal (figure 1-6), with the direction of 
the Z axis common to all three. No piezoelectric 
effects are directly associated with the Z axis, and 
an electric field applied in this direction produces 
no piezoelectric deformation in the crystal, nor will 
a mechanical stress along the Z axis produce a 
difference of potential. Because the growth lines 


Section I 

Physical Characteristics of Piezoelectric Crystals 

are generally missing, optical effects are usually 
employed to locate the Z azis in raw quartz. (See 
paragraphs 1-121 to 1-124.) Quartz properties are 
such that light waves passing through a crystal 
are effectively divided into two rectilinear com- 
ponents, with one component traveling faster than 
the other except when the light ray is directed 
parallel to the Z axis. The optical effects are found 
to be symmetric about the Z axis, and thus whereas 
optical instruments may be used to determine this 
axis, they cannot be used to distinguish an X from 
a Y axis. For this reason the Z axis is commonly 
designated as the optic axis. The optical effects 
associated with the propagation of polarized light 
parallel to the optic axis not only are used to locate 
the Z axis in unfaced quartz (crystals, such as 
river quartz, whose natural faces have been de- 
stroyed), but to identify left from right quartz, 
and to locate twinned regions. Plane polarized light 
traveling parallel to the optic axis will be rotated 
in one direction or the other according to whether 
the crystal is left or right. To an observer looking 
toward the light source the rotation will be clock- 
wise for right-handed quartz and counterclockwise 
for left-handed quartz, with the amount of rota- 
tion depending upon the wavelength, being greater 
for blue light (short wavelength) and less for red 
light (long wavelength). Since the crystal lattice 
along the optic axis has no properties that distin- 
guish one direction from the other, the choice of 
the -1-Z and the — Z reference directions are en- 
tirely arbitrary for either right or left crystals. 


'^2 



Figure 1-6. XY plane of quartz showing three sets 
of rectangular axes; X,Y,Z, X^Y^Z, X.,YjZ (Z axis is 
perpendicular to plane of paper) 


WADC TR 56-156 


11 



Section i 

Physical Characteristics of Piezoelectric Crystals 

Y Axis 

1-53. The Y axes are chosen at right angles to the 
Z axis and to the growth lines of the m faces. See 
figure 1-4. For either left or right quartz, the 
positive end of a Y axis emerges from an m face 
that is adjoined by a z face at the end selected as 
the +Z direction. The Y axes are generally called 
the mechanical axes in contradistinction to the X 
axes, which are called the electrical axes. These 
names originated from the fact that simple com- 
pressional and tensional mechanical stresses along 
either an X axis or a Y axis would cause a polariza- 
tion of the X axis, but not of the Y axis. The names 
are somewhat misleading, for polarization in the 

Y direction is also possible if a crystal undergoes 
shearing or flexural strains. In practice, the Y 
axis of a quartz stone is usually determined after 
the Z and X axes have been located. 

X Axis 

1-54. The X axes are parallel to the growth lines 
of the m faces, and to the lines bisecting the 120- 
degree prism angles. The positive end of an X axis 
is the direction that forms a right-handed coordi- 
nate system (see figure 1-4) with the Y and Z axes. 
This makes the directional sense of the X axis in 
right quartz the reverse, rather than the mirror 
image, of that in left quartz. Thus, in right quartz 
the negative ends of the X axes emerge from the 
prism corners that lie between the x faces, where- 
as, in left quartz the positive ends emerge from 
these .corners. 

1-55. In either right or left quartz when the X 
axis undergoes a tensional strain (stretching), a 
positive charge appears at the end emerging be- 
tween the X faces; and when the X axis is com- 
pressed, this end becomes negatively charged. The 
X axis of raw quartz is usually determined by 
optical and x-ray methods. See paragraphs 1-126 
and 1-127. 

THEORY OF PIEZOELECTRICITY 

Report Announcing Discovery of the 
Piezoelectric Effect 

1-56. The theory of the cause of piezoelectricity 
stated in the most general terms is substantially 
the same today as it was at the time of its dis- 
covery. The following is the original report by 
Pierre Curie on the piezoelectric effect, which in- 
cludes a statement of the theory that led to its 
discovery. The paper was read at the April 8, 1880, 
meeting of the societe mineralogique de France, 
and is recorded in the Bulletin, soc. min. de France, 
volume 3, 1880. 


1-57. “Crystals which have one or more axes 
whose ends are unlike, that is to say, hemihedral 
crystals with inclined faces, have a special physical 
property, that they exhibit two electric poles of 
opposite names at the ends of those axes when they 
undergo a change of temperature : this is the phe- 
nomenon known as pyroelectricity. 

1-58. “We have found a new way to develop elec- 
tric polarization in crystals of this sort, which con- 
sists of subjecting them to different pressures 
along their hemihedral axes. 

1-59. “The effects produced are analogous to those 
caused by heat: during a compression, the ends 
of the axis along which we are acting are charged 
with opposite electricities; when the crystal is 
brought back to the neutral state and the com- 
pression is relieved, the phenomenon occurs again, 
but with the signs reversed; the end which was 
positively charged by compression becomes nega- 
tive when the compression is removed and re- 
ciprocally. 

1-60. “To make an experiment we cut two faces 
parallel to each other, and perpendicular to a hemi- 
hedral axis, in the substance which we wish to 
study ; we cover these faces with two sheets of tin 
which are insulated on their outer sides by two 
sheets of hard rubber; when the whole thing is 
placed between the jaws of a vise, for example, 
we can exert pressure on the two cut surfaces, that 
is to say, along the hemihedral axis itself. To per- 
ceive the electrification we used a Thomson elec- 
trometer. We may show the difference of potential 
between the ends by connecting each sheet of tin 
with two of the sectors of the instrument while 
the needle is charged with a known sort of elec- 
tricity. We may also recognize each of the elec- 
tricities separately; to do this we connect one of 
the tin sheets with the earth, the other with the 
needle, and we charge the two pairs of sectors from 
a battery. 

1-61. “Although we have not yet undertaken the 
study of the laws of this phenomenon, we are able 
to say that the characteristics which it exhibits are 
identical with those of pyroelectricity, as they have 
been described by Gaugain in his beautiful work 
on tourmaline. 

1-62. “We have made a comparative study of the 
two ways of developing electric polarization in a 
series of non-conducting substances, hemihedral 
with inclined faces, which includes almost all those 
which are known as pyroelectric. 

1-63. “The action of heat has been studied by the 
process indicated by M. Friedel, a process which 
is very convenient. 


WADC TR 56-156 


12 



1-64. “Our experiments have been made on blende, 
sodium chlorate, boracite, tourmaline, quartz, cala- 
mine, topaz, tartaric acid (right handed), sugar, 
and Seignette’s salt. 

1-65. “In all these crystals the effects produced by 
compression are in the same sense as those pro- 
duced by cooling; those which result from reliev- 
ing the pressure are in the same sense as those 
which come from heating. 

1-66. “There is here an evident relation which 
allows us to refer the phenomena in both cases to 
the same cause and to bring them under the fol- 
lowing statement : 

1-67. “Whatever may be the determining cause, 
whenever a hemihedral crystal with inclined faces, 
which is also a non-conductor, contracts, electric 
poles are formed in a certain sense ; whenever the 
crystal expands, the electricities are separated in 
the opposite sense. 

1-68. “If this way of looking at the matter is cor- 
rect, the effects arising from compression ought 
to be in the same sense as those resulting from 
heating in a substance which has a negative co- 
efficient of dilation along the hemihedral axis.’’ 

Asymmetrical Displacement of Ciiar9e 

1-69. The atomic lattice of piezoelectric crystals is 
assumed to consist of rows of alternating centers 
of positive and negative charges so arranged that 
the structure as a whole has no center of sym- 
metry. When such a lattice undergoes a deforma- 
tion, a displacement will result between the 
“centers of gravity’’ of the positive and negative 
charges. It is this displacement that results in a 
net unneutralized dipole moment, the polarity of 







^ 

® © 

1 

® 

4 - 

-f ^ 

1 ® © j 


i 





POPLARITY UNDER NEUTRALIZED POLARITY UNDER 

COMPRESSIONAL DIPOLE TENSlONAL 

• STRAIN STRAIN 


Figure 1-7. Effective polarities resulting from sudden 
displacements of the centers of charge of a neutral- 
ized dipole. (If after displacement, the crystal were 
maintained indefinitely in the strained position, the 
effective polarity would eventually be neutralized by 
an accumulation of ions at the poles. A sudden return 
from such a state would thus result in an effective 
polarization in the unstrained position! 


Section I 

Physical Characteristics of Piezoelectric Crystals 

which depends upon the previous equilibrium posi- 
tions of the positive and negative centers of charge 
and the direction of the displacement, as indicated 
in figure 1-7. 

1-70. In the case of a crystal with a center of sym- 
metry, a uniform strain in the crystal will always 
result in as much displacement of like charges in 
one direction as in another, and hence there will 
be no net shift of the centers of opposite charge 
relative to each other. A distribution of charges 
having a center of symmetry is illustrated in figure 
1-8. Note that the centers of charge, both positive 
and negative, are at the geometrical center. If a 
uniform stress — compressional, or shearing — is 
applied along any axis, it can be seen that the 
center of either type of charge will at all times 
remain undisturbed, and thus the net piezoelectric 
effect will be null. 

1-71. Lord Kelvin was the first to propose a molec- 
ular model with a charge distribution designed to 
explain the physical and electrical characteristics 
of alpha quartz. See figure 1-9. This model was 
accepted generally by the crystallographers until 
the theory failed to conform to X-ray tests. When 
a beam of X-rays enters a crystal, the intersecting 
atomic planes can be likened to partialh’^ silvered 
mirrors, each passing part of the beam, but reflect- 
ing the rest. Since the distance between adjacent 
parallel planes is on the order of an X-ray wave- 
length, the waves reflected from adjacent planes 
will tend to alternately annul and reinforce each 
other as the angles of incidence vary. The inter- 
ference pattern oh a photographic film will show 
an array of spots indicating the angles at which 
the reflected waves from different planes arrive 
in phase. From such data, with the X-ray wave- 
length known, it is possible to determine the rela- 
tive orientation of, atomic planes, and hence to 
reconstruct the arrangement of the atoms in the 
crystal. The X-ray data on alpha quartz reveals a 



1 

%) 

1 

% 

1 



Figure 1-8. Example of distribution of charges 
having a cenfer of symmetry * 


WAOC TR 56-156 


13 






Section i 

Physical Characteristics of Piezoelectric Crystals 


\ _ /K _ / 


\ @-G) ®-Q 


STQ. 

/ \ / \ 


/ \ / N 


/ 

/®~®\ 
/ \ / ^ 


©-© ©-©^ 

' ^ \ 

,©-a \ 


^©-©^ 


Vtn y^' 'y->w y— v' 


©-© 

/ \ 


®— © 

/ \ 


Figure 1-9. Kelvin's molecular model of the charge 
distribution of alpha quartz. The positive direction 
shown for the X axis corresponds to that of right 
quartz, for a compression of the crystal along that 
axis will cause the piezoelectric polarities to coincide 
in sign with the X-axial directions. For left quartz, 
the sign of the Y, as well as the X, axis, must be 
reversed in order to maintain a right-handed 
coordinate system * 



ARRANGEMENT OF ATOMS IN ALPHA QUARTZ, VIEWED ALONG AN X AXIS 


Figure 1-10> Arrangement of atoms In alpha quartz. 
Plane of paper corresponds to YZ plane In crystal 




Figure 1-11. Equivalent distribution of charges that account for observed piezoelectric effects of alpha quartz. 
(A) Piezoelectric polarity along X axis of right quartz due to compression along Y axis. (8) Piezoelectric polarity 
of Y axis of right quartz due to shearing stress, where the resultant strain is equivalent to a compression along 
the axis designated GH. (Note that in both A and 8, the piezoelectric effect is due to a rocking of the axial 
dipoles, and not to their compression or extension. To achieve the same deformations by the converse effect, 
equal voltages, but opposite in sign to the polarizations indicated, are applied across the respective axes)* 


WADC TR 56-156 


14 



more complex structure than was once suspected. 
See figure 1-10. Nevertheless, the early crystal 
model, as postulated by Lord Kelvin, still can be 
accepted as an approximation if we treat a single 
one of his molecules as representing simply the 
equivalent charge distribution within the lattice, 
as indicated in figure 1-11. 

1-72. Figure 1-11 A shows the displacement occur- 
ring when a compressional stress is applied along 
the Y axis, or a tensional stress is applied along 
the X axis of a right-handed crystal. Note that the 
center of positive charge shifts in the negative 
direction of the X axis, and that the center of nega- 
tive charge shifts in the positive direction; how- 
ever, there is no net displacement along the Y axis. 
If the direction of the stress is reversed, so also is 
the effective polarity. 

1-73. The polarization of the Y axis due to a shear- 
ing strain is illustrated in figure 1-llB. Assume 
that vectors A and B represent a simple shearing 
stress applied at right angles to the Y axis. If A 
and B are equal and opposite forces, there will be 
no displacement of the center of mass; however, 
since these forces are not directed in the same 
straight line, they create a couple which would 
maintain a rotational acceleration about the center 
of mass unless opposed by an equal and opposite 
couple. This counter-couple is represented by vec- 
tors C and D. If now, the forces are combined 
vectorially, they may be represented as a longi- 
tudinal tension in the EF direction, or as a longi- 
tudinal compression in the GH direction. Consider 
the charge displacement from the point of view of 
a GH compression. Note that each of the positive 
charges is forced to shift slightly in the -1-Y direc- 
tion, whereas each of the negative charges is dis- 
placed in the — Y direction. The net separation of 
the centers of charge thus causes the Y axis to 
become positively polarized at its geometrically 
positive end, and negatively polarized at its geo- 
metrically negative end. 

1-74. Since the compression can be further ana- 
lyzed into two components of equal magnitude — 
one horizontal, and the other vertical — it can be 
seen (figure 1-llA) that the polarities which these 
would induce along the X axis tend to cancel (for 
reinforcement to occur, one of the rectangular 
components would need to be tensional and the 
other compressional) , and hence little or no polari- 
zation will appear in this direction. 

MODES OF VIBRATION 

1-75. If a piezoelectric crystal is suddenly released 
from a strained position, the inertia and elasticity 


Section I 

Physical Characteristics of Piezoelectric Crystals 

of the crystal will tend to maintain a state of 
mechanical oscillation of constant frequency about 
one or more nodal points, lines, or planes of equi- 
librium, and alternating voltages will appear ac- 
cording to the particular mode of vibration. These 
are called the normal, or free, vibrations of a 
crystal, as distinct from the forced vibrations due 
to applied alternating mechanical or electrical 
forces that may differ in frequency from the crys- 
tal’s natural resonance. The normal vibrations 
may, in turn, be of two general types; the free- 
free and the clamped-free vibrations. Free-free 
vibrations ai-e those which would occur if a vibrat- 
ing crystal were floating in empty space, where, 
regardless of the particular mode, the center of 
gravity is a nodal point. Clamped-free vibrations 
are those that would occur if a crystal Were 
clamped at some point, or points, thereby prevent- 
ing all normal modes except those at which nodes 
occur at the clamped points. For example, in a free- 
free vibration the ends of the crystal are free to 
move; however, if these ends are clamped, the 
resonant vibrations must be such that the ends 
become nodes. However, if a crystal is clamped 
only at those points which would be nodes in a 
free-free vibration, in the ideal case no interfer- 
ence results, and the resonance is still that of a 
free-free mode. 

1-76. There are three general modes of vibration 
for which quartz crystal units are commercially 
designed: extensional, shear, and flexure. Funda- 
mental vibrations of each of these modes are illus- 
trated in figure 1-12. Higher harmonics up to and 
including the fifth are also widely used. Harmonic 
vibrations higher than the seventh have special 
high-frequency applications, but are rarely em- 
ployed commercially. 

1-77. A variation of the shear vibration is the 
torsional mode, which is readily excited in cylin- 
drical crystals; however, except for laboratory 


T^i 



( B ) I D> 


Figure 1-12. Useful fundamental modes of quartz 
plates. (A) Flexural. (B) Extensional or longitudinal. 
(C) Face (or length-width) shear. (D) Thickness shear.* 


WADC TR 56-156 


15 




Sactien i 

Physical Characteristics of Piezoeiactric Crystals 

measurements of the properties of solids and 
liquids this mode is not in general use. 

Frequency of Quarts Vibrations 

1-78. The frequencies of the normal mechanical 
oscillations of a quartz plate may be considered as 
those at which standing waves will be established 
by reflection from the crystal boundaries. The posi- 
tions of the nodes of the standing waves are pre- 
determined by the geometry of the crystal, and by 
any difference that may exist in the velocities of 
propagation for the different wave components. 
The wavelength of a particular mode (but not the 
wave shape, if the velocity of one component dif- 
fers from that of another) conforms only to the 
dimensions of the crystal faces. The frequency is 
related to the wavelength by the equation : 

f = I 1-78 (1) 

where: v = velocity of propagation 
X = wavelength 

The fundamental equation of the velocity of propa- 
gation is: 


1—78 (2) 

where: c = stiffness factor in the direction of propa- 
gation 

p = density 



where: s = - = elastic compliance factor in the di- 
^ rection of propagation 

Length- (or Width-) Extensienal Mode 

1-79. The motion of the atoms in an extensional 
mode is parallel to the direction of propagation. In 
the case of rectangular plates, stationary waves 
are established in the length direction by the inter- 
ference of reflections from the opposite ends, 
where the wavelength is given by the formula : 



X = 


n 


1—79 (1) 


where 1 is the length and n is an integer (1, 2, 3, 
etc.) equal to the harmonic. Thus, the frequency 


of a length-extensional mode is : 

f = ^ 1-79 (2) 

or, as expressed in terms of a frequency constant: 
f = ^ 1—79 (3) 

where: kj = ^ = frequency constant 

for length-extensional mode 

This formula, as well as the similar formulas for 
the shear and flexure modes, can be used to indi- 
cate the approximate dimensions required for a 
particular frequency when the appropriate fre- 
quency constant is known. Although the velocity 
of propagation decreases somewhat as the fre- 
quency increases, because of an increase in the 
frictional losses, this decrease is negligible for 
most purposes, and the same frequency constants 
that hold for the fundamental are also valid for 
the first few overtones. However, because of the 
coupling that exists between the length-extensional 
mode and other modes, the effective value of k will 
vary with changes in the w/1 (width/length) 
ratio. Equation (3) also applies to width-exten- 
sional modes except that 1 is replaced by the 
width, w. 

Thlcknasi-Extensionai Mod* 

1-80. This mode is little used today because of the 
close coupling that exists between it and the over- 
tones of other modes. It is a mode that can be 
excited in a crystal whose thickness dimension is 
parallel to the electrical (X) axis (X-cut crystal) 
— the vibrations being such that the crystal alter- 
nately becomes thicker and thinner. Formerly, 
when X-cut crystals were widely used, the same 
crystal was often employed for the control of 
either a high- or a low-frequency circuit — using 
the thickness-extensional mode for the former and 
the length-extensional mode for the latter. Today, 
however, the more stable thickness-shear mode has 
almost entirely replaced the thickness-extensional 
mode in high-frequency circuits. The thickness- 
extensional frequency is given by the formula: 


nv _ nkj 

2t “ ~r 


1—80 (1) 


where v is the velocity of propagation in the thick- 
ness direction, n is the harmonic (n = 1, 3, 6,— for 
practical cases, although even harmonics of very 
small intensities have been observed), and kj is 


WADC TR 56-156 


16 



the generalized frequency constant. Actually, the 
effective thickness, t, decreases somewhat for the 
overtones, so that the correct value of kj is slightly 
greater for the harmonics than for the funda- 
mental. 

Thickaest-Shear Mode 

1-81. The motion of the atoms in a thickness-shear 
mode is parallel to the major (length-width) faces 
of the crystal, whereas the wave propagation is 
parallel to the thickness dimension. The equation 
for the fundamental frequency when the thickness 
is very small compared with the length and 
width is: 


f = J 1-81 (1) 

where: v = velocity of propagation along thickness 
dimension 
t = thickness 

or, as expressed in terms of a thickness-shear fre- 
quency constant: 

f = ^ 1—81 (2) 

The thickness-shear is also called the “high- 
frequency shear” in contradistinction to the “face,” 
“length-width,” or “low-frequency” shear. The 
overtones of the thickness-shear mode may have 
components (reversals of phase) in the length and 
width directions as well as along the thickness. 
The more general formula for the frequency is: 



where m, n, and p are integers representing the 
harmonic component in the t, 1, and w directions, 
respectively. The above equation applies to an iso- 
tropic medium; however, since the elastic con- 
stants in quartz are not the same in all directions, 
the thickness-shear formula has been modified to : 


f ~ 1^3 W d" ni j3 "t" n*' 


(P - D* 


w* 


1-81 (4) 

where ai and a, are constants to be determined 
empirically. For most applications, t is much 
smaller than 1 and w, and n = p = 1, so that the 

formula, f = sufficiently accurate. 

Fac»>Shcar Med* 

1-82. The face-shear mode involves a more com- 


WAOC TR 56-156 


17 


Section I 

Physical Characteristics of Piezoelectric Crystals 

plex relation among the crystal dimensions. A 
complication arises from the fact that the wave is 
effectively divided into two components— one 
propagated along the length, and the other along 
the width. Each of these separate components has 
its own series of possible harmonics, so that the 
resultant frequencies of the face-shear modes are 
not necessarily integral multiples of the funda- 
mental. The approximate-frequency equation is : 





^2 „2 

m n 

-p-+ 


ki 




m^ 


+ ar 


w^ 


1—82 (1) 

where m and n are integers representing the length 
and width harmonics, respectively. The symbol a, 
is a constant of proportionality, approximately 
equal to one, which is inserted when the velocity 
of propagation along w is not the same as that 
along 1. If the face of the plate is square, the 
formula for the fundamental frequency is reduced 
to approximately: 

f = — 1—82 (2) 

w 

where: Ic* = k^-\/2 

The fundamental vibration, where m = n = 1, is 
shown in figure 1-12C. Note that the shape of the 
deformation is not that of a parallelogram, as it 
would be if the plate were slowly compressed along 
a diagonal. Rather, the vibrational distortion is a 
dynamic one, and the resultant wave must be in 
the same phase at all points. Figure 1-13 repre- 
sents the face-shear mode for m = 6, n =; 3. Note 
that the number of nodes in each row is equal to 
m, and the number in each column is equal to n. 



figurm 1-13. faco-ehaar modi* for m = 6, n = 3. 
Dots inditato itodot * 



Section I 

Physical CharactorisHcs of Piezooloctric Crystals 

Longth-Width-Flexnral Mod* 

1-83. The length-width-flexural mode is a bending 
of the crystal in the length-width plane. Normally, 
the crystal is so mounted that the ends are free to 
vibrate in a free-free mode. The formula for the 
frequency involves the root of a transcendental 
equation, but expressed in tenns of a frequency 
constant, the equation becomes: 

f = ^ 1-83 (1) 

The convenience of a common frequency constant 
for all practicable harmonics is not realized in the 
case of length-width flexures, where the “constant” 
ks is a function not only of the particular harmonic, 
but also of 1 and w. However, for long, thin rods 

^^less than 0.1, where n is the harmonic) k, is 

approximately independent of the dimensions, and 
fixed values of k, can be assumed for the particular 
harmonics of different types of cuts. Because of 
the elastic cross constants in quartz, which relate 
a field in one direction to a polarization in a per- 
pendicular direction, a flexure may be accompanied 
by a torsion. To prevent this, the length of a crys- 
tal to be operated in a flexural mode should lie 
somewhere in a YZ plane. 

Length-Thickncts-Flexural Med* 

1-84. Length-thickness flexures are used to control 
frequencies in the audio range. To obtain this 
mode, two long, thin plates of the same cut are 
cemented together with the electrical axes opposed, 
so that, when an alternating voltage is applied 
across the outer faces, one crystal strip expands as 
the other contracts, and vice versa — the over-all 
effect being a flexural vibration. The normal fre- 
quency of a free-free length-thickness flexure is 
given by an equation similar to that for the length- 
width flexure, except that the thickness, t, is sub- 
stituted for the width, w. Thus: 

f = nka (p) 1-84 (1) 

Frequency Ranqe of Normal Modes 

1-85. Standard quartz crystal units are designed 
for frequencies from 400 cycles to 125 megacycles 
per second. Laboratory devices have employed 
thickness flexure crystals for the control of fre- 
quencies as low as 50 cycles per second, and, by 
exciting the higher thickness-shear modes, control 
of frequencies higher than 200 megacycles per 
second have been realized. At these high frequen- 

WADC TR 56-156 18 


cies, however, so many interlocking modes are 
possible that it is difficult to prevent a crystal from 
jumping from one mode to another during slight 
variations of temperature, unless a very precise 
fabrication of the crystal unit has been achieved. 
The high-frequency limit of the lower harmonics is 
reached when the dimensions are so small that 
either the crystal cannot be driven without the 
risk of shattering, or that the impedances intro- 
duced by the mounting become proportionately too 
large for practicable operation. 

1-86. The practical frequency ranges of the differ- 
ent modes are as follows : 

Flexure Mode — 

Length-thickness : 0.4 to 10 kc 
Length-width: 10 to 100 kc 
Extensional Mode — 

Length : 40 to 850 kc 
Thickness : 500 to 15,000 kc 
Shear Mode — 

Face : 100 to 1800 kc 

Thickness (fundamental) : 500 to 20,00Q kc 
Thickness (overtones) : 16,000 to 125,000 kc 

ORIENTATION OF CRYSTAL CUTS 

Riqht-Haiidad Caardinatu Sysfam 

1-87. With the positive sense of the quartz X, Y, 
and Z axes determined as in paragraphs 1-52, 1-53, 
and 1-64, the positive sense of rotation about the 
axes is fixed by the conventions of a right-handed 


z 



Figure I -14. Positive directions of angles of rotation 
according to conventions of right-handed 
coordinate system 



coordinate system for both right and left quartz. 
If one imagines a' right-handed screw pointing 
towards the positive end of an axis of rotation, as 
represented in figure 1-14, the direction of an angle 
of rotation is considered positive if the rotation 
advances the screw in a positive direction — ^this 
corresponds to a clockwise rotation if observed 
when looking towards the positive end of the axis 
of rotation. The reverse, or counterclockwise, 
angles of rotation are taken as negative. The sense 
of the axes are such that the angles of rotation are 
positive when the directions of rotation are from 
-f-X to +Y, -f-Y to -{-Z, and -fZ to -f X. The axial 
and rotational conventions permit a particular cut 
of crystal to have the same rotation symbol for 
both right and left quartz. 

Refotiea Symbols 

1-88. To specify the orientation of a piezoid cut, 
the following system, as recommended by the 
I. R. E. in 1949 is in general use. The crystal blank 
to be described is assumed to have a hypothetical 
initial position, with one corner at the origin of 
the coordinate system, and the thickness, length, 
and width lying in the directions of the rectangular 
axes. There are six possible initial positions, each 
of which is specified by two letters, the first letter 
indicating the thickness axis, and the second letter 
indicating the length axis. These positions are thus 
designated xy, xz, yx, yz, zx, and zy. The xy and yx 
positions are shown in figures 1-15 and 1-16, re- 
spectively. The starting position is so chosen that 
the final orientation may be reached with a mini- 
mum number of rotations. These rotations are 
taken successively about axes that parallel the 


z 



figun 1-15 xy Initial poshlon tor designating 
oriontation of crystal cwf 

WADC TR 55-156 19 


SncHon I 

Physical Charadnristics of Pinsanlnclflc Crystals 

dimensions of the crystal at the time of rotation. 
Only the first rotational axis will coincide with a 
rectangular axis; however, the positive direction 
of any axis of rotation is that defined by the XYZ 
system for the initial position. A single rotation is 
sufficient for describing the majority of standard 
cuts, and three rotations is the maximum in any 
case. The dimensions and axes of rotation are indi- 
cated by the symbols, t, 1, and w, for thickness, 
length, and width, respectively. The Greek letters 
<f>, 0, and ^ designate the first, second, and third 
angles of. rotation, respectively. The following ex- 
ample, illustrated in figure 1-17, is a complete geo- 
metrical specification of a crystal plate : 

yztwl 30V15'’/25° 
t = 0.80 ± 0.01 mm 
1 = 40.0 ± 0.1 mm 
w = 9.00 ± 0.03 mm 

The lettered combination at the beginning of the 
specification is called the “rotation symbol.” The 
first two letters, yz, of the symbol indicate the 
initial position, and the next three letters, twl, 
state the axes of rotation and the order in which 
the rotations are taken. The three angles, all posi- 
tive in this case, give the orientation and are listed 
in the same order as the respective rotations. The 
dimensions listed are those of the particular plate, 
and are not to be considered as necessary specifi- 
cations for that type of cut. For circular plates, 
the initial position will indicate which directions 
are to be considered thickness and length, so that 
the same rotation ssmibol is used as for rectangular 


z 



Figure 1-16 yx Initial pogitlon for designating 
oriontation of crystal tut 


Section I 

Standard Quartz Elements 



Figure 1-17 Orientation of crystal having the rotational specIHcations ystwl: 30°/15°/25'’ 


plates: in specifying the dimensions, however, 1 
and w are replaced by the diameter. 

PIEZOELECTRIC ELEMENTS 

1-89. The performance characteristics of a crystal 
plate are dependent on both the particular cut and 


the mode of vibration. For convenience, each "cut- 
mode” combination is considered a separate “pie- 
zoelectric element,” and the more commonly used 
elements have been assigned a letter sjnnbol. For 
example, the thickness-shear mode of the AT cut 
is designated as element A. 


STANDARD QUARTZ ELEMENTS 

1-90. The principal quartz elements are given be- symbols listed first, 
low, with those which have been assigned element 


Element 

Symbol 

Name of 
Cut 

Rotation Symbol and 
Orientation 

Mode of Vibration 

Frequency 
Range in KC 

A 

AT 

yxl 35“21' or yzw 35'’21' 

thickness-shear 

600 to 125,000 

B 

BT or YT* 

yxl — 49®8' or yzw — 49®8' 

thickness-shear 

1,000 to 76,000 

C 

CT 

yxl 37°40' or yzw 37°40' 

face-shear 

300 to 1,100 

D 

DT 

yxl — 52‘’30' or yzw — 52*30' 

face-shear ' 

60 to 500 

E 

-!-6“X 

xyt 5* 

length-extensional 

60 to 600 


* The YT cut, which is essentially the same as the BT cut, was developed independently by Yoda in Japan. 

WADC TR 56-156 20 




Section I 

Standard Quartz Eiomonts 


Element 

Symbol 

Name of 
Cut 

Rotation Symbol and 
Orientation 

Mode of Vibration 

Frequency 
Range in KC 

F 

-18.5°X 

xyt -18.5° 

length-extensional 

60 to 300 

G 

GT 

yxlt -51°7.5745° 

width-extensional 

100 to 550 

H 

5°X 

yxt 5° 

length-width flexure 

10 to 50 

J 

Duplex 5°X 

xyt 5° (right quartz) and xyt 5° 
(left quartz) 

length-thickness flexure 

0.4 to 10 

M 

MT 

xytl 0° to 8.5°/±34° to ±50° 

length-extensional 

50 to 500 

N 

NT 

xytl 0° to 8.5°/±38° to ±70° 

length-width flexure 

4 to 100 

— 

AC 

yxl 31° or yzw 31° 

thickness-shear 

1,000 to 15,000 

— 

BC 

yxl —60° or yzw —60° 

thickness-shear 

1,000 to 20,000 

— 

ET 

yxl 66°30' or yzw 66°30' 

combination flexure 
and face-shear 

600 to 1,800 

— 

FT 

yxl —57° or yzw —57° 

combination flexure 
and face-shear 

150 to 1,500 

— 

V 

xzlw or xywl 15° to 29°/— 14° 
to -54° and 13° to 29°/27° to 42° 

thickness-shear 

1,000 to 20,000 
(fundamental) 

— 

V 

xzlw or xywl 0° to 30°/±46° to 
±70° 

face-shear 

60 to 1,000 

— 

X 

xy 

length-extensional 

40 to 350 

— 

X 

xz 

width-extensional 

125 to 400 

— 

X 

xy or xz 

thickness-extensional 

350 to 20,000 

— 

Y 

yx or yz 

thickness-shear 

500 to 20,000 


TYPES OF CUTS 

1-91. The standard quartz elements can be divided 
into two groups : in the first group belong those 
crystals which are most conveniently described as 
being rotated X-cut crystals, and in the second 
group belong those crystals which are most con- 
veniently described as being rotated Y-cut crystals. 
The first will hereafter be designated as the X 
group, and the second as the Y group. 

1-92. The X and Y cuts have their thickness di- 
mensions parallel to the X and Y axes, respectively, 
with the length and width dimensions parallel to 
the two remaining axes. See figure 1-18. Thus, in 
describing a crystal orientation, the X cut is the 
equivalent of the two initial positions xy and xz, 
and the Y cut is represented by the initial positions 
yx and yz. Belonging to the X and Y groups, then, 
are those crystals whose rotation symbols begin 
with the letters x and y, respectively. As a general 
rule, from the X group, the low-frequency crystal 
units are obtained, and from the Y group, the 
medium- and high-frequency units. A third group 


of crystals is theoretically possible, where the 
initial position is a Z cut (thickness parallel to the 
Z axis) : however, because the piezoelectric effect 


Z AXIS 



Plgur 0 1-78 Orientation of X, Y, and Z cut plates 


WADC TR 56-156 


21 



Section I 

Standard Quartz Element* 


is restricted to the X and Y axes, the electrodes 
must be placed across one of these axes, which for 
the Z cut, would be at the edges — not a convenient 
location. Nor have other cuts, more or less simply 
oriented relative to a Z cut, been found to have 
optimum performance characteristics. However, 
there are experimental Z cuts, such as some of the 
ring-shaped crystals, which have proven of high 
quality, even though not practical for general use. 

The X Group 

1-93. The principal crystals of the X group are 
listed below with the frequency ranges for which 
they have found commercial application : 

Name of Cuts Frequency Range in KC 


X 

40 to 20,000 

5°X 

0.9 to 500 

-18°X 

60 to 350 

MT 

50 to 100 

NT 

4 to 50 

V 

60 to 20,000 


z 



(B) 5*XCUT 


Figure 1-19 shows the orientations of an xy initial 
position (X cut with the length parallel to the Y 
axis) for the various cu^'-. 



X 


(O-ie'x CUT 


z' z 



X 


(0)MT CUT 


Ptgun 1-19. The X group. (Tho second rotations ot the MT, NT, and V cuts are shown only for the positive angies) * 
WADC TR 56-156 22 





Section I 

Standard Quartz Element* 

THE X CUT 

1-94. The X cut was the original quartz plate in- 
vestigated by Curie, and thus is sometimes called 
the “Curie cut.” This cut was also the first to be 
used as a transducer of ultrasonic waves and as the 
control element of radio-frequency oscillators. 
However, because of its comparatively large co- 
efficient of temperature, the X-cut plate is now 
rarely used in radio oscillators. As a transducer of 
electrical to mechanical vibrations, especially at 
high frequencies of narrow bandwidth, the X cut 
has a high electromechanical coupling efficiency, 
and is still widely used to produce ultrasonic waves 
in gases, liquids, and solids. These applications are 
largely for testing purposes, such as the measure- 
ment of physical constants and the detection of 
flaws in metal castings. 

1-95. CHARACTERISTICS OF X-CUT PLATES 
IN THICKNESS-EXTEN SIGNAL MODE 

Description of Element: X cut; xy or xz; thick- 
ness-extensional mode. 

Frequency Range: 350 — 20,000 kc (fundamental 
vibration) ; lower frequencies when coupled 
as transducer for generating vibrations in 
liquids and solids. 

nk 

Frequency Equation: f = — ^ (n = 1, 3, 5, . . .) 

Frequency Constant: k., = 2870 kc-mm. 
Temperature Coefficient: 20 to 25 parts per million 
per degree centigrade; negative (i.e. for each 
degree increase or decrease in temperature, 
the frequency respectively decreases or in- 
creases 20 to 25 cycles for each megacycle of 
the initial frequency — a rise in temperature 
of 10°G would thus cause the frequency of a 
5000-kc crystal to drop 1000 to 1250 cycles 
per second.) 

*Methods of Mounting: Sandwich and undamped 
air-gap — for oscillator circuit; transducer 
mounting depends upon particular type of 
mechanical load. 

Advantages: Mechanical stability, economy of cut, 
efficiency of conversion of electrical to me- 
chanical energy, and large frequency constant 
make this piezoelectric element preferred for 
the radiation of high-frequency acoustic 
waves when the ratio of the highest to the 
lowest frequency need not exceed 1.1. 
Disadvantages: Large temperature coefficient, 
tendency to jump from one mode to another, 
and the difficulty of clamping crystal in a 
fixed position without greatly damping the 

* See paragraphs 1-132 to 1-171. 

WADC TR 56-156 



RATIO OF OPTICAL TO MECHANICAL AXIS 

Figure 1-20. Frequency constant for length-exten- 
sional mode (curve A) of X-eut crystal where the 
width and length are parallel to the Z and Y axes, 
respectively. Curve B is the frequency constant of a 
face-shear mode coupled to a second flexural mode, 
whose Interference makes the crystal useless for w/l 
ratios between 0.2 and 0.3, unless the thickness 
approaches the dimensions of the width * 

normal vibration prevent this element from 
being preferred for oscillator control. An 
electromechanical coupling factor of 0.096, 
which is only one-fourth that of the best syn- 
thetic crystals, makes this element inefficient 
as a radiator of a wide band of frequencies. 

1-96. CHARACTERISTICS OF X-CUT PLATES 
IN LENGTH-EXTENSION AL MODE 

Description of Element; X-cut; xy; length-exten- 
sional mode. 

Frequency Range: 40—360 kc. 

21 Ic 

Frequency Equntion: f = -y-’ (n = 1, 2, 3, . . .) 

Frequency Constant: Varies with w/l ratio— see 
figure 1-20. 

Temperature Coefficient: Negative**, varies with 
w/l ratio — see figure 1-21 ; zero coefficient if 
w/I = 0.272 and w = t. 

Methods of Mounting: Sandwich, air gap, wire, 
knife-edge clamp, pressure pins, cantilever 
clamp; more than one pair of electrodes re- 
quired for overtones ; transducer mounting de- 
pends upon particular type of mechanical load. 

** All quartz bars have negative temperature coeffieienta 
for pure length-extensional vibrations, although a zero 
coefficient is obtainable for certain cuts. 


24 


Section i 

Standard Quartz , Clemenis 



W/Jl 

Figure 1-21. Temperature coefficient for length-ex- 

tensional mode of X-euf crystal, where w is parallel 
to the Z axis, and t = 0.051 

Advantages: For w/1 ratios from 0.35 to 1.0, the 
fundamental length-extensional vibration is 
not strongly coupled to other modes, and hence 
the resonance is easily excited and of good 
stability except for drift during temperature 
variations. Although not preferred over zero- 
temperature-coefficient cuts, this element, 
with temperature control, is reliable for use 
in low-frequency oscillators, and for long, 
thin bars, for use in filters. However, its most 
important application is to produce ultrasonic 
vibration in gases, liquids, and solids, when 
the ratio of highest to lowest frequency need 
not exceed 1.1. 

Disadvantages: Inefficient as transducer of any 
but narrow frequency band, since electro- 
mechanical coupling is only one-fourth that of 
the better synthetic crystals. Strong coupling 
with a flexural mode makes the crystal use- 
less at w/l ratios between 0.2 and 0.3 (see 
figure 1-20) , and a weak coupling with a shear 
mode causes the frequency constant to de- 
crease as the w/l ratio approaches 1.0. This 
coupling to other modes interferes with the 
frequency response of the element when used 
in filters, unless the w/l ratio is 0.1 or less. 
Although for long thin bars the temperature 
coefficient is only about 2 parts per million per 
degree, this is greater than the minimum ob- 
tainable with fi^X-cut bars. 

1-97. CHARACTERISTICS OF X-CUT PLATES 
IN WIDTH-EXTENSION AL MODE 

Description of Element: X-cut; xz; width-ex- 
tensional mode. 

Frequency Range: 126 to 400 kc. 

nlCi 

Frequency Equation: f = — ^ (n = 1, 2, 3, . . .) 

Frequency Constant: Varies with w/l ratio; see 

WADC TR 56-156 



Figure 1-22. Frequency characteristics of X-cut crystal 
vibrating in width-extensional mode, where the width 
is parallel to the Y axis, w/l ratios not included 
between the two outer curves will have interfering 
modes. K, is in kc-inches 

figure 1-22, which shows the face dimensions 
that will have a single frequency near the de- 
sired resonance. Plates with dimensions not 
included between the two outer curves will 
have interfering modes. 

Temperature Coefficient: Negative ; approximately 
10 parts per million per degree centigrade, but 
varies with w/l ratio. 

Methods of Mounting: Sandwich, air gap, wire, 
knife-edge clamp, pressure pins, cantilever 
clamp. 

Advantages: If cut with dimensions within the 
single-frequency range shown in figure 1-22, 
this element can be used in temperature-con- 
trolled low-frequency oscillators and narrow- 
band-pass filters. With the thickness dimen- 
sion ground for a particular high frequency, 
the same crystal unit may be used to generate 
either of two widely separate frequencies. 
Disadvantages: Relatively large temperature co- 
efficient prevents this element from being pre- 
ferred over the low-coefficient cuts. 


25 




Section I 

Standard Quartz Elements 

THE 5° X CUTS 


284 


1-98. The 5°X cut is the orientation that provides 
a zero temperature coefficient for the lengthwise 
vibrations of long, thin X bars, as shown in figure 
1-23. Thus, this cut is preferred over the non- 
rotated X cut for use in low-frequency filters and 
control devices. Its length-extensional, length- 
width-flexural, and duplex length-thickness-flex- 
ural modes are defined as the elements E, H, and 
J, respectively ; the last named element, J, provid- 
ing the lowest frequencies. However, the 6°X ele- 
ments are also coupled to the other modes, so that 
for w/1 ratios much greater than 0.1 the frequency 
spectrum is little improved over that of the length- 
extensional mode of the X cut. Furthermore, as the 
w/1 ratio increases, so also does the temperature 
coefficient. For these reasons the 5°X elements are 
especially advantageous only when the w/1 ratio 
is 0.1 or less. These long, thin bars are used com- 
mercially for the control of low-frequency oscilla- 
tors and as filters, and are particularly adaptable 
for use in telephone carrier systems. 

1-99. CHARACTERISTICS OF ELEMENT E 

Description of Element: 5°X cut; xyt: 6° ; length- 
extensional mode. 

Frequency Range: 50 to 500 kc. 

nlc 

Frequency Equation: f — (n = 1, 2, 3, . . .) 

Frequency Constant: Varies with w/1 ratio (see 
figure 1-24 )’. 

Temperature Coefficient: Varies with w/1 ratio 
(see figure 1-25, which holds for temperatures 
between 45 and 55 degrees centigrade). The 



-40 -30 -20 -10 0 10 20 30 40 90 

ANGLE OF ROTATION AROUND THE X AXIS 


Figure 1-23. Temperature eoefUeieM for length-ex- 
tensional mode of long, thin X-group bars versus 
angle of rotation * 



Figure 1-24. FrequeiKy constant versus w/1 ratio 
for element E * 

frequency deviation of representative E ele- 
ments of different w/1 ratios is shown in fig- 
ure 1-26, where the initial frequency is taken 
at 26°C. 

Note that the temperature coefficient in parts 
per hundred per degree is the slope of a curve, 
and varies from positive to zero to negative 
as the temperature increases. 

Methods of Mounting: Wire, knife-edge clamp, 
pressure pins, cantilever clamp; more than 
one pair of electrodes required for overtones. 

Advantages: The low temperature coefficient and 
a large ratio of stored mechanical to electrical 
energy make this element 4 )ref erred for filter 
networks. Long, thin bars have only a very 



O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 
RATIO OF WIDTH TO LENGTH 


Figure 1-25. Temperature coefhclent versus w/1 ratio 
for element E at temperatures between 45° and 55° C* 


WAOC TR 56-156 


26 



Saction. I 

Standard Qvartx Elamantt 




fjgwra 1-28. Tomparature coefflWant varsws w/l rath 
for a/amant H * 


Figaro 1~26. Poreonlego froquottcy dovlathn for B 
olomonti of vartovi w/l ratios. Initial 
tmmporaturo — 25°C 

weak coupling to other modes and are used 
for both filter networks and low-frequency 
oscillators. If a w/l ratio greater than 0.15 is 
desired, a ratio of approximately 0.39 is opti- 
mum insofar as a low temperature coefficient 
is concerned. 

Disadvantages: At v//l ratios between 0.2 and 0.3 
the length-extensional mode is so closely 
coupled to the length-width flexure that the 
crystal is useless; as the width is increased 
the coupling of the length-extensional to the 
face-shear mode becomes stronger, and the 
temperature coefficient becomes larger. How- 
ever, because of the large electro-mechanical 
coupling of this element, w/l ratios of 0.35 to 
0.5 can still be favorably used in filters if a 
temperature coefficient less than 4 parts per 
million is not required. 

1-100. CHARACTERISTICS OF ELEMENT H 

Description of Element: 5°X cut; xyt; 5° ; length- 
width flexure mode. 

Frequency Range: 10 to 100 kc. 



Figaro 1-27. Froqaoney constant vorsus w/t ratio 
for olomont H 


Frequency Equation: f = nksW/1’, (n = 1, 2, 3, 

...) 

Frequency Constant: Varies with w/1 ratio (see 
figure 1-27). 

Temperature Coefficient: Varies with w/l ratio 
(see figure 1-28). 

Methods of Mounting: Wire, in vacuum; free-free' 
flexures of long, thin bars have nodal points 
for the fundamental vibration at a distance 
of 0.224 X 1 from the ends; two electrically 
opposite pairs of electrodes are plated on each 
side of the YZ faces, with “ears” at the nodal 
points for soldering to the mounting wires. 
See figure 1-29. When the polarity of the lower 
electrodes causes a contraction of the bar, the 
polarity of the upper electrodes causes an ex- 
tension, and vice versa — the over-all result 
being a flexural deformation. 

Advantages: For long, thin bars the length-width 
flexural mode is resonant at much lower fre- 
quencies than is the length-extensional mode. 
This advantage, combined with the favorable 
electro-mechanical coupling, and reasonably 
low temperature coefficient, has made this ele- 
ment useful in very-low-frequency filters 
where only a single frequency is to be selected. 
When mounted in vacuum, a Q of 30,000 is 
obtainable. 



Figaro 1-29. Blomont H, showing division of oloctrodo 
plating for oxciting furtdamontal rnodo. Similarly 
divided oloctrodos are on reverse side. Tho nodal 
"oars" whom tho mounting wins are attachod, am 
at a distanco of approximatoly 0.224 times tho longth 
from tho ends * 


WAOC TR 56-156 


27 


FREQUENCY CONSTANT IN KILOCYCLE-CENTIMETERS 


Section I 

Standard Quartz Elements 




Figure 1-30. Frequency constant versus t/l 
ratio for element J 




RATIO m/i 


Figure ?-3l. Percentage frequency deviation for J 
elements. The smaller t/l ratio is representative of a 
1.2-kc element, and the larger t/l ratio Is repre~ 
sentative of a lO-kc element 


Figure 1-32. Frequency constant versus w/l ratio for 
various resonances of — 18° X-cut crystal. A Is the 
width-extensional mode. B is the face-shear mode, 
which, at small w/l ratios, is strongly coupled to D, 
the second flexural mode. C represents the band 
between the antiresonant (upper curve) and the res- 
onant (lower curve) frequencies of the length-exten- 
sional mode of element F. Note the weak coupling 
between C and D-B * 


WADC TR 56-156 


28 


Section I 

Stondard Quortx Eiementa 


Disadvantages: The ratio of stored mechanical to 
electrical energy is not as large as that of the 
length-extensional mode, and because of this, 
the element does not give as broad a band-pass 
spectrum. Also, the elfect of the shear stresses 
causes the temperature coefficient to become 
highly negative as the w/1 ratio is increased. 
Finally, the damping effect of the air is 
greater for flexural than for other vibrations, 
so that flexure crystals should be mounted 
only in evacuated containers. 

1-101. CHARACTERISTICS OF ELEMENT J 

Description of Element: Duplex 5°X cut; xyt: 5° 
(right quartz), and xyt: 5° (left quartz); 
length-thickness flexure mode. 

Frequency Range: 0.4 to 10 kc. 

Frequency Equation: f = nk,t/l* (n = 1, 2, 3, 

.. .) 

Frequency Constant: Varies with t/1 ratio (see 
figure 1-30). 

Temperature Coefficient: Varies with both the t/1 
ratio and the temperature ; figure 1-31 shows 
the total relative frequency deviation of two 
elements of different t/1 ratios, the initial fre- 
quencies being those at 26°C. The temperature 
coefficients in parts per hundred at a given 
temperature are the slopes of the curves at 
that point. Note that the temperature at which 
a zero coefficient is obtained increases as the 
t/1 ratio decreases. At temperatures below 
that of a zero-coefficient point, the coefficient 
is positive ; at temperatures above, it is nega- 
tive. 

Methods of Mounting: Headed-wire, in vacuum; 
two thin plates are cemented together with 
polarities opposed so that only one pair of 
electrodes, plated on opposite YZ faces, are 
required ; the crystal element is supported at 
the nodal points, which for the fundamental 
vibration are at a distance 0.224 x 1 from 
each end. 

Advantages: Small temperature coefficient and 
low resonant frequencies (among the lowest 
obtainable with quartz) make this element 
useful in providing stable control for sonic- 
frequency oscillators, and as a component of 
single-frequency filters. 

Disadvantages: Not economical for control of fre- 
quencies above 10 kc. 

1-102. CHARACTERISTICS OF ELEMENT F 

Description of Element: — 18.5°X cut; xyt: 
—18.5° ; length-extensional mode. 

Frequency Range: 60 to 300 kc. 


tik 

Frequency Equation: f = -j-^ (n = 1, 2, 3, . . .) 

Frequency Constant: Varies slightly with w/1 
ratio (see figure 1-32). 

Temperature Coefficient: 25 parts per million per 
degree centigrade — varies very little with 
changes in the w/1 ratio. 

Methods of Mounting: Wire, knife-edge clamp, 
pressure pins, cantilever clamp ; more than 
one pair of electrodes required for overtones. 

Advantages: The extremely weak coupling of this 
element to the face-shear and second flexure 
modes, represented by curves B and D, re- 
spectively, in figure 1-32, permits a better 
frequency spectrum than can be obtained with 
element E for w/1 ratios greater than 0.1. For 
this reason, the F element used to be pre- 
ferred over the E element as a filter plate, 
and was the principal quartz element in the 
channel filters of coaxial telephone systems. 
This is no longer true because channel filters 
now use -j-5°X plates which are smaller and 
conserve quartz. 

Disadvantages: Relatively large temperature co- 
efficient prevents this element from being pre- 
ferred for oscillator control or as a channel 
filter if wide variations in temperature are to 
be expected. Also, the F plate is larger than 
the E plate of the same frequency and thus 
consumes more quartz. 

1-103. CHARACTERISTICS OF ELEMENT M 

Description of Element: MT cut; xytl : 0° to 8.5°/ 
±34° to ±50°; length-extensional mode. 

Frequency Range: 50 to 500 kc. 

Tik 

Frequency Equation: i — — p (n = 1, 2, 3, . . .) 

Frequency Constant: Varies with w/1 ratio and 
angles of rotation (see figure 1-33) . 

Temperature Coefficient: Varies with w/1 ratio 
and angles of rotation (see figure 1-34), and 
with the temperature. The total relative fre- 
quency deviation of an 8.5°/±34° M element, 
where the initial frequency is taken at 40°C, 
is shown in figure 1-35. Note that the tem- 
perature coefficient, which is the slope of the 
curve, changes from positive to negative as 
the temperature increases, with the zero co- 
efficient occurring at 63°C. 

Methods of Mounting: Wire, knife-edge clamp, 
pressure pins, cantilever clamp; more than 
one pair of electrodes required for overtones. 

Advantages: The MT crystals were developed in 
an effort to overcome the large negative tem- 
perature coefficients of the X-cut and the 5°X- 


WADC TR 56-156 


29 



Section I 

Standard Quarts Elomontt 


cut length-extensional modes for the larger 
w/1 ratios. See figures 1-21 and 1-26. The un- 
favorable temperature characteristics are 
caused by the coupling of the extensional to 
the face-shear mode, the latter having a high 
negative temperature coefficient. However, if 
the crystal is rotated about its length, an 
orientation will be found where the face-shear 
mode has a zero temperature coefficient ; that 
is, the coefficient will pass from negative to 
positive values. The low temperature coeffi- 
cient of the length dimension will thus be pre- 
served even though the coupling to the shear- 
mode has not, itself, been diminished. The 
low temperature coefficient makes the M ele- 
ment advantageous for oscillator control in 
the 50-to-100 kc range, and for use in narrow 
band filters, such as pilot-channel filters in 
carrier systems, where wide temperature 
ranges are to be encountered. The 8.5'’/34® 
rotation with a w/1 ratio of approximately 
0.42 provides the greatest electromechanical 
coupling of the M elements, and hence the 
broadcast bandpass of the MT cut for use in 
filters. 

Disadvantages: The electromechanical coupling 
rapidly decreases as the w/1 ratio increases, 
so that at ratios greater than 0.7 the element 
is too selective for filter use, and of too small 
a piezoelectric activity to be advantageous for 
oscillator control. Maximum electromechani- 
cal coupling is obtained with w/1 ratios of 
0.39 to 0.42 ; but for a maximum bandwidth 
the E element is preferred. Although the in- 
terference of the face-shear temperature co- 
efficient is reduced, the coupling to that mode 
remains relatively strong ; so where the tem- 
perature varies very little, or where the sec- 
ondary frequency effects are undesirable, the 
F element is preferred. 

1-104. CHARACTERISTICS OF ELEMENT N 

Description of Element: NT cut; xytl: 0° to 8.5”/ 
±38° to ±70”; length-width flexure mode. 

Frequency Range: 4 to 100 kc. 

Frequency Equation: f (n = 1, 2, 3, . . .) 

Frequency Constant: Varies with w/1 ratio (see 
figure 1-36). 

Temperature Coefficient: For w/1 ratios of 0.2 to 
0.5, low coefficients are obtained by double 
rotations of 0” to -|-8.5”/±50”. Typical fre- 
quency deviation curves are shown in figure 
1-37, where the initial temperature is taken 
at 25°C. Note that a zero temperature coeffi- 



RATtO OF WIDTH TO ceNOTH 


Figure 1-33. Frequency constant versus w/1 rath for 
M olomonts having low temperature coeFRelents. C is 
the curve of the most commonly used MT orientation * 


UJ 



Figure 1-34. Temperature coefficient versus w/1 ratio 
for M elements * 

dent occurs at approximately 10”C. To pro- 
duce a zero temperature coefficient at 25°C 
for w/1 ratios of 0.05, the angles of rotation 
should be as shown in figure 1<^8. 

Methods of Mounting: Wire, in vacuum; special 
characteristics are the same as for the H ele- 
ment. See paragraph 1-100. 


WADC TR 56-156 


30 





S«cMon I 

Standard Quarts ElamanI* 



TEMPERATURE IN DEGREES CENTIGRADE 


figure 1-35, Freqwency-tomperatwre eharac- 
t»ri$ties of ahmant M * 





TEMPERATURE CO 


Figura 1-37. Fraquancy-tamparatura eharactarlstiea of alamant 
N. Tha largar w/l ratio Is typical of 100-kc alamanis, and tha 
smallar w/l ratio la typical of 16-kc alamants 


Figura 1-36. Fraquancy constant varsus 
w/l ratio for alamant N 



Figura 1-38. Anglos of rotation for N ala- 
mant with a w/l ratio of 0.05 which will 
provida xaro tamparatura coafficiaat 
at 35°C * 


Advantages: The principal advantage of the N 
element is that the second rotation reduces 
the temperature coefficient for the flexure vi- 
bration of long, thin crystals. This is accom- 
plished by changing the width from near 
parallelism to the Z axis to near parallelism 
to the X axis. Theoretically the ideal rotation 
would be 90°, except that the piezoelectric 
effect would be reduced to zero. As a compro- 
mise, secondary rotations, about the length, 
of 39° to 70° are made. Besides reducing the 
flexure-mode temperature coefficient of the 
long, thin crystals, the rotation also reduces 
the negative coefficient for the shear modes 
at the higher w/l ratios, as in the case of the 
M element. Where wide temperature ranges 


must be met, this element is preferred for 
very-low frequency oscillators, and ‘in single- 
frequency filter selectors. As the control ele- 
ment of an oscillator, it can maintain the 
frequency within ±0.0026% over a normal 
room-temperature range without temperature 
control. 

Disadvantages: The electromechanical coupling is 
rather weak, more so for the larger than for 
the smaller w/l ratio. As a consequence, the 
bandwidth is too narrow for the element to 
be used as a band-pass filter of communica- 
tion channels, and the piezoelectric activity is 
so low that special circuits are required for 
its use in oscillators. 


WAOC TR 56-156 


31 



Section I 

Standard Quartz Elomantz 

THE V CUT 

1-105. The V cut, developed by S. A. Bokovoy and 
C. F. Baldwin of RCA, is actually an entire series 
of cuts obtained by a sequence of double rotations 
of an initial X-cut plate. The first rotation angle, 
0 , is taken about the Z azis, and the second rota- 
tion angle, 6, is taken about the Y' axis (the di- 
mension of the crystal that is initially parallel to 
the Y axis). For each angle there is an angle 9 
at which the crystal will have a given temperature 
coefficient for a particular mode of vibration. Nor- 
mally, the combination of angles desired is one 
that will provide a zero temperature coefficient; 
however, it may be that a small positive or nega- 
tive coefficient is required to counterbalance an 
opposite temperature coefficient inherent in the 
external circuit to which the crystal is to be con- 
nected. For this purpose curves of $ plotted against 
<t>. are shown in figures 1-40 to 1-41 for small posi- 
tive and negative temperature coefficients, as well 
as for a zero temperature coefficient. Other ^ and 
6 combinations may be extrapolated to give tem- 
perature coefficients differing from the actual 
values shown. It should be noted that when the 
rotation about the Z axis is equal to ±30°, the 
thickness dimension becomes parallel to a Y axis, 
and hence the crystal is in the position of the Y 
cut, with the Y' axis coinciding with an X axis. 
Thus, if <^ = ±30°, the V cut is essentially the 
same as a rotated Y cut, and in this case would 
embrace practically the entire Y family. On the 
other hand, if 4 . = 0°, the V cut becomes simply 
a singly rotated X cut — but with rotations about 
the Y axis, not the X axis as in the case of the 
5°X and the — 18°X cuts. However, when <f> = 0°, 
the V cut does overlap the MT and NT cuts. 

1-106. CHARACTERISTICS OF V-CUT 

PLATES IN THICKNESS-SHEAR MODE 

Description of Flement: V cut; xslw or X 3 rwl: 15° 
to 29°/-14° to -54° and 13° to 29°/27° to 
42° (see temperature coefficient curves in fig- 
ure 1-40 for exact <i> and 9 combinations) ; 
thickness-shear mode. 

Frequency Range: 1000 to 20,000 kc (funda- 
mental) ; higher frequencies on overtones. 

Frequency Equation: f = (fundamental vibra- 
tion when t << 1 and w). Figure 1-39 shows 
the frequency constant of the zero-tempera- 
ture-coefficient series of V cuts as a function 
of the first rotation angle. The upper curve, 
designated k, ( -f e) , applies to positive angles 
of 6, the second rotation, whereas the lower 


curve, designated k, (—6), applies to nega- 
tive angles of 9. 

Temperature Coefficient: Figure 1-40 shows the 
combinations of ^ with positive values of 9 
that provide temperature coefficients of -|-15, 
0, and —15 parts per million per degree centi- 
grade, and those combinations of ^ with nega- 
tive values of 9 that provide temperature 
coefficients of -f 5, 0, and —5 parts per mil- 
lion per degree centigrade. 

Methods of Mounting: Sandwich, air gap, clamped 
air-gap, button. 

Advantages: The principal advantage of the V cut 
is that a given temperature coefficient may 
be obtained from a large choice of orienta- 
tions, and with a minimum in trial-and-error 
procedure. Not only can a series of zero- 
coefficient plates be obtained, but also plates 
with coefficients of desired sign and magni- 
tude for annulling the known frequency-tem- 



Ftgun 1-39. Frmgueney constant versus (angle of 
rotation about Z axis) for the thickness-shear mode of 
V-eut crystals when 9, the second angle of rotation, 
is so chosen that a zero temperature coefficient Is 
obtained. The upper and lower curves are for positive 
and negative values of 9, respectively 


WADC TR 56-156 


32 


Saction I 

Standcmi Quarts EUmant* 


perature effects of the circuits in which the 
plates are to be used. The V cut is the only 
member of the X group that provides a zero 
temperature coefficient for high-frequency 
vibrations ; and because of the large choice of 
rotation angles, one or the other of the V 
orientations will frequently permit the maxi- 
mum use of an unfaced or badly twinned 
mother crystal. Because their larger fre- 
quency constants permit a thicker and less 
fragile crystal, the orientations with a nega- 
tive 6 are preferred for the higher frequen- 
cies. Also, small deviations in negative values 
of 6 produce less variation in the temperature 
coefficient than do the same deviations in posi- 
tive values of 6 . Hence, the negative orienta- 
tions of 6 are also generally more dependable 
for obtaining a desired temperature coeffi- 
cient. On the other hand, positive values of 6 
permit a less bulky crystal for the lower fre- 
quencies, a less critical frequency constant, 
less interference from spurious frequencies, 
and for accurately determined orientations, a 
broader temperature deviation for a given 
deviation in frequency. At <^ = 30°, the values 
of = —49°, -t-31°, and -|-35°3r are substan- 



flgur* 1-40. kmtatlona of 0 to for tMcknoss-sfcoar 
mod* of V eat, which prwido tho tomporaturo 
coofficionts imfieotod for ooch curve 


tially the same as the BT, AC, and AT cuts, 
respectively, of the Y group, as described in 
paragraphs 1-114, 1-111, and 1-112. The. chief 
use of the thickness-mode V cut is for the 
control of high-frequency oscillators. 

Disadvantages: The possibility of spurious fre- 
quencies close to the desired fundamental is 
the most troublesome limitation of the V cut 
operating in a thickness-shear mode. As a 
general rule, the coupling between the desired 
and the stray modes diminishes as the initial 
rotation <t> is increased. At values of less 
than 13°, the interference is too great for sta- 
ble operation. Because of the relatively poor 
frequency spectrum, the V cut is not readily 
adaptable for use in selective networks. With 
a certain amount of cut-and-try experimenta- 
tion, the more objectionable modes may be 
reduced by grinding down the width and 
length dimensions. For angles of 4 , close to 
30° the length and width dimensions most 
important to avoid are approximately the 
same as those given in paragraphs 1-112 and 
1-114 for the AT and BT cuts, respectively. 

1-107. CHARACTERISTICS OF V-CUT 
PLATES IN FACE-SHEAR MODE 

Description of Element: V cut; xzlw or xywl: 0° 
to 30°/±45° to ±70° (see temperature co- 
efficient curves in figure 1-41 for exact 4 and 
6 combinations) ; face-shear mode. 

Frequency Range: 60 to 1000 kc. 

Frequency Equation: f = k,/w (fundamental for 
square plates). 

Frequency Constant: Insufficient data exist to plot 
the curve of k« for all the combinations of 
and 6 corresponding to this element. However, 
in the case of the zero-coefficient plates, as 
the positive value of 6 approaches 37.5°, k, 
approaches 3070 kc-mm, and as the negative 
value of 9 approaches —62.5°, ki approaches 
2070 kc-mm. 

Temperature Coefficient: Figure 1-41 shows the 
combinations of 4> and 9 that provide tem- 
perature coefficients of -f 6, 0, and —5 parts 
per million per degree centigrade. 

Methods of Mounting: Wire, cantilever clamp. 

Advantages: The principal advantage is the low 
temperature coefficient, which makes the ele- 
ment useful for low-frequency oscillators and 
filters. The large choice of orientation angles 
is also advantageous for obtaining the maxi- 
mum number of cuts from a given mother 
crystal, particularly if the presence of twin- 
ning or other defects limit the dimensions in 


WAOC TR 56-156 


33 



section I 

Standard Quartx ElomanH 



Figure 1-41. Relations of $ to if>, for fate-shear 
mode of V cut, which provide temperature coeRIcients 
of 0, -|-5, and — 5 parts, per million per 
degree centigrade 

the directions at which rough bars would 
normally be cut. Also, the angles for small 
predetermined positive and negative coeffi- 
cients permit a crystal to be cut which can 
exactly annul the known temperature effects 
of the external circuit. As indicated in figure 
1-41, small deviations in the orientations 
angles will cause minimum deviations in the 
temperature coefficient when 0 = 0° to 15°, 
and $ is negative. On the other hand, maxi- 
mum piezoelectric activity is obtained when 
<t> is large, and 0 is positive. As a general rule, 
the positive values of 6 are used for the higher' 
frequencies and the negative values of 6 for 
the lower frequencies. The zero-temperature 
cuts for <t> = 30° are substantially the same 
as the CT and DT cuts of the Y gproup. See 
paragraphs 1-116 and 1-116, respectively. 
Disadvantages: Care must be taken that flexure 
modes are not strongly coupled to the face- 
shear mode. Such coupling may be reduced 
by making the plates square, or nearly so. For 
angles of approaching 30°, the thickness 
should be approximately within the limits 
given for the C and D elements in paragraphs 
1-115 and 1-116. 


Tka Y Grovp 

1-108. The principal crystals of the Y group are 
listed below with the frequency range for which 
they have found commercial application : 


Hame of Cut 

Frequency Range in KC 

Y 

1000 to 

20,000 

AC 

1000 to 

15,000 

AT 

500 to 100,000 

BC 

1000 to 

20,000 

BT 

1000 to 

75,000 

CT 

300 to 

1100 

DT 

60 to 

600 

ET 

600 to 

1800 

FT 

150 to 

1500 

GT 

100 to 

660 


Figure 1-42 shows the orientations of a yx initial 
position (Y cut with the length parallel to the X 
axis) for the various cuts. In special cases the 
width may be parallel to the X axis, but this is 
the exception rather than the rule, unless the plate 
is square or circular. With the exception of the GT 
cut, the crystals of the Y group are used in their 
shear modes — face shear for the low-frequency 
elements, and thickness shear for the high-fre- 
quency elements. The Y cut, itself, has a large 
positive temperature coefficient; and, because of 
coupling between the thickness-shear mode and 
the overtones of the face-shear mode, it also ex- 
hibits sharp irregularities in its frequency spec- 
trum. However, by rotation about the X axis, zero 
temperature coefficients may be obtained, and the 
coupling between the shear modes can be greatly 
diminished. This coupling becomes zero at the 
angles of the AC and BC cuts, and the frequency 
constant of the thickness-shear mode has minimum 
and maximum values, respectively, for these two 
orientations. Figure 1-43 shows the thickness- 
shear frequency constant, and figure 1-44 the 
thickness-shear temperature coefficient, with each 
plotted as a function of the angle of rotation. For 
the face-shear mode, the frequency constant and 
the temperature coefficient are shown in figures 
1-45 and 1-46, respectively, plotted as functions of 
the angles of rotation. 

THE Y CUT 

1-109. The Y cut was introduced commercially in 
the late 1920’s, at which time its principal advan- 
tage was that it could be clamped at its edges, 
whereas the X cut would not oscillate if the edge 
movement were even slightly restricted. The use 
of a Y cut, vibrating in a shear-mode, was origi- 
nally suggested by E. D. Tillyer of the American 
Optical Company, to whom a U. S. patent was 


WADC TR 56-156 


34 








SMtion I 

Stondard Quaiix Elvmvnt* 



Figure 1-42. Rotation anglos of Y cut about X axis which provide the principal members of the Y group. The 
GT cut is the only member having a second rotation (±45° about the Y' axis). The *'S" indicates that 
the positive end of the X axis points toward the observer 


issued in 1933. For this reason, the Y cut is some- 
times called the Tillyer cut. For several years this 
crystal was used extensively in commercial and 
military transmitters mounted in mobile equip- 
ment, and also in commercial broadcast transmit- 
ters where the Y cut's readily excited oscillations 
permitted the use of crystal oscillators with low 
plate voltages. However, due to the strong cou- 
pling between the thickness-shear and the over- 
tones of the face-shear and flexure modes, the Y 
cut’s frequency spectrum is very poor. Also, small 
irregularities in the dimensions of the crystal 
readily produce abrupt changes in the frequency. 


A typical frequency-temperature curve of a Y-cut 
crystal is shown in figure 1-47. Today, the Y cut 
has been almost entirely replaced by the rotated 
cuts having small temperature coefficients, and 
the Y cut’s only major application now is that of 
transducer for generating shear vibrations in 
solids. 

1-110. CHARACTERISTICS OF Y-CUT 

PLATES IN THICKNESS-SHEAR MODE 

Description of Element: Y cut; yx or yz; thick- 
ness-shear mode. 

Fre(/uency Range: 500 to 20,000 kc; much lower 


WADC TR 56-156 


35 



Sectipn I 

Standard Quartz Elamants 



ROTATION ABOUT X AXIS - -Wi f »•+ ROTATION ABOUT X AXIS 

IN DEGREES IN DEGREES 


Figure 1-43. Frequency constant versus angle of rotation about X axis for thickness-shear elements of Y group. 
(Values shown for Y, BT, and BC cuts are smaller than the average) * 



so -TS -60 -4S -30 -IS 0 IS 30 4S 60 7S BO 

ANGLE OF ROTATION ABOUT X AXIS IN DEGREES ( t ) 


Figure 1-44. Temperature coefficient versus angle of rotation about X axis for 
thickness-shear elements of Y group * 


WADC TR 56-156 


36 




Section I 

Standard Quarts Eloinonts 


> 

K 


it 3E»0 
•-8 

iu 300 0 

^ u 

>2 ZTSO 

8:i 

,.i tsoo 

z ' 

Su 1210. 








15 


- 

a 









n 










■ 

K 

m 








Gi 


IS 

m 

■ 





3 

Si 

Wo 2000 
c o 

tk .J 



IS 

22 

m 

■ 






n 

►0 -7S -so -AS -30 -IS 0 IS 30 45 SO TS »0 


“ ORIENTATION ANCLE IN DEGREES 1^) 

Figurm 1-45. FnqtMHcy constant vorsut attgh of rota- 
tion about X axis for faco-$hoar olomontt of Y group 

frequencies when bonded to solids for use as 
transducer. 

Frequency Equation: f (fundamental vibra- 
tion) . 

Frequency Constant: k, = 1981 kc-mm (average 
value) . 

Temperature CoeffUient: Varies with dimensions 
of crystal and with temperature but is usually 
between 76 and 125 parts per million per de- 
gree centigrade, and is positive, with an aver- 
age value of 86 parts per million per degree 
centigrade. 

Methods of Mounting: Sandwich ; air gap, clamped 
air gap ; bonded to solids when used as trans- 
ducer. 

Advantages: Ratio of stored mechanical to elec- 
trical energy is larger than that of any other 



TEMPERATURE 


Figure 1-47. Tomporaturo-froquohey charactorislics 
typical of tho Y-eut,, thkknoss-shoar olomont. Yho 
froquoney jumps aro most apt to occur whan small 
discropauclos aro prossat In tbo thkknoss-cllmonsioH 



Figure 1-46, Tomporaturo coofHclont versus angle of 
rotation about X axis tor face-shear elements of 
the Y group 


quartz element; this large ratio, combined 
with the quartz crystal’s superior strength, 
makes the Y cut desirable as a generator of 
shear vibrations in solids for the purpose of 
measuring or testing the solids’ physical 
properties. This element is the easiest of all 
quartz cuts to excite into vibration, and thus 
requires the lowest voltages for operation. 
Large temperature coefficient makes element 
useful as a sensitive detector of variations in 
temperature. 

Disadvantages: Large temperature coefficient, dis- 
continuities of resonant frequencies, and poor 
frequency spectrum make this element a sec- 
ondary choice for use in either oscillator or 
filter circuits. Special Y cuts, such as the 
block- and doughnut-shaped crystals in figure 
1-48, vibrate in a combination mode com- 
posed of coupled shear and flexure modes, 
and have zero temperature coefficients at cer- 
tain temperatures. However, because of the 
prevalence of spurious frequencies, the large 
volume of quartz used per cut, and the diffi- 
culties of mounting, these crystals have little 
practical use. 



Figure 1-48. Y-cut block and doughnut-shaped crys- 
tals which can provide zero temperature coefBclents 
for certain combination modes 


WADC TR 56-156 


37 





Section I 

Standard Quartz Elomontt 


1-111. CHARACTERISTICS OF AC-CUT 

PLATES IN THICKNESS-SHEAR MODE 

Description of Element; AC cut; yxl: 31° ; length- 
thickness-shear mode. 

Frequency Range: 1000 to 15,000 kc (fundamental 
vibration) . 

Frequency Equation: f = (fundamental vibra- 

V 

tion when t < < 1 and w) . 

Frequency Constant: k, = 1656 kc-mm. 

Temperature Coefficient; 20 parts per million per 
degree centigrade; positive. 

Methods of Mounting: Sandwich, air gap, clamped 
air gap, button. 

Advantages: This element vibrates in a very pure 
length-thickness mode with an excellent fre- 
quency spectrum. It has the lowest frequency 
constant of all the quartz thickness modes 
and thus permits a smaller thickness, and 
hence a more economical cut, for use at the 
low end of the high-frequency spectrum. For 
a given temperature, the electrical parame- 
ters of an AC crystal unit can be predeter- 
mined with an accuracy equal to, or greater 
than, that of the more commonly used AT 
units. 

Disadvantages: The principal disadvantages of 
the AC cut is its relatively large temperature 
coefficient; because of this the element has 
found little commercial use, and the low-co- 
efficient AT cut, with, an orientation close 
enough to that of the AC for the coupling be- 
tween the shear modes to be small, is generally 
preferred. 

1-112. CHARACTERISTICS OF ELEMENT A 

Description of Element: AT cut; yxl: 36°21'; 
length-thickness-shear modes; or, }rzw: 36° 
21'; width-thickness-shear mode. 


Frequency Range: 500 to 1000 kc (special cuts) ; 
1000 to 15,000 kc (fundamental vibration) ; 
10,000 to 100,000 kc (overtone modes). 
Frequency Equation: 

f = ks/t (fundamental vibration when t< <1 and w) 


f 



+ ajf 


(P- 1) 
w* 


2 


where m, n, and p are integers. 


Frequency Constant: k, = 1660 kc-mm. 

Temperature Coefficient: 0.0 at 26°C; figure 1-49 
shows the total relative frequency deviation 
for the normal maximum, minimum, and aver- 
age angles of this element; the temperature 
coefficient at each point on a curve is the slope 
at that point in parts per hundred. At 
^ = 36°16', the temperature coefficient will 
vanish at 45°C, changing from negative to 
positive as the temperature increases. Opti- 
mum orientations for zero coefficients at other 
temperatures are given below : 


Deg. C 

♦ 

20 

35°18' 

20 

35°27' (overtones) 

75 

35°31' 

76 

35°33' (overtones) 

85 

85°83' 

86 

36°86' (overtones) 

100 

36’36' 

190 (max.) 

S6°26' 


Methods of Mounting: Sandwich, air gap, clamped 
air gap, button. 

Advantages: The excellent temperature-frequency 
characteristics make this element preferred 
for high-frequency oscillator control wher- 
ever wide variations of temperature are to 
be encountered; it is particularly applicable 



-60 40 to 0 . to 40 60 60 *100 

TCMPCRATURE (*0) 

Figure 1-49. Temperafure-froquency tharactorlstles of o/omont A 


38 


WADC TR 56-156 


for aircraft radio equipment where sharp 
changes in tonperature may be frequent, but 
where the added weight of constant'tempera- 
ture ovens is undesirable. The angle of orien- 
tation is sufficiently close to that of the AC 
cut for the coupling between the shear modes 
to be weak, so that the resonant frequency 
can be isolated from that of other modes, ex- 
cept for certain dimensions of w and 1. The 
A element also shares the low frequency con- 
stant of the AC element, and this is preferred 
for frequencies at the low end of the high- 
frequency spectrum. However, because of its 
superior temperature-frequency and piezo- 
electric characteristics as compared with the 
BT characteristics, the A element may well 
be preferred for the control of frequencies in 
the vhf range, even though the BT cut has 
the larger frequency constant. 

Disadvantages: Because of the large thickness di- 
mensions that would be required, the A ele- 
ment is generally not economical for the gen- 
eration of frequencies below 1000 kc, although 
special circular cuts have been used at fre- 
quencies as low as 500 kc. In its normal high- 
frequency range, the most troublesome prob- 
lem is to find the proper length and width 
dimensions which will permit the desired fre- 
quency to be widely separated from other 
modes. Although the orientation of the AT 
cut is close to that of the AC cut, there still 
exists a fair amount of coupling to the face- 
shear modes, and to the extensional and flex- 
ural modes along the X axis. Sufficient infor- 
mation is not available to avoid a certain 
amount of trial and error in grinding and fin- 
ishing an AT blank to provide an optimum 
frequency spectrum at a desired frequency; 
however, there are certain X and Z' (the di- 
mension parallel to the Z axis before rotation) 
values that can be avoided by use of the fre- 
quency equations which hold approximately 
for the less complex of the unwanted modes. 
The following equations give the face dimen- 
sions of an AT cut which will produce un- 
wanted resonances at the same frequency, f, 
as the thickness-shear mode. 


For extensional modes along X: 


X = 


2488 n 
f 


(n = 1, 3, 5, . . .) 


For flexure modes along X : 


X=: 


1338.4 n 
f 


(n = 2, 4, 6, 



Section I 

Standard Quartz Elements 


For shear modes along X ; 


V 2542.0 n 


(n = l, 8, 5,...) 


For shear modes along Z': 


Z'= 


2540.0 n 


(nrri, 3, 5, ...) 


With f expressed in kc, X and Z' are given in 
mm. Either X or Z' may be the length, with 
the other dimension being the width. It has 
been found that the unwanted modes are 
somewhat restricted by giving the plate a con- 
vex contour, and also by the use of circular 
plates. The convex contour is possible for all 
but the very thin plates that are used at fre- 
quencies above 15,000 kc. A 1000-kc crystal 
may have a contour of 3 to 5 microns. The 
equations above hold for flat plates, and be- 
come increasingly in error as the contour is 
increased. 


1-113. CHARACTERISTICS OF BC-CUT 

PLATES IN THICKNESS-SHEAR MODE 

Description of Element: BC cut; yxl: —60°; 
length-thickness-shear mode. 

Frequency Range: 1000 to 20,000 kc (fundamental 
vibration). 

k 

Frequency Equation: ^ ~ (fundamental vibra- 
tion when t < < 1 and w) . 

Frequency Constant: kj = 2611 kc-mm. 

Temperature Coefficient: 20 parts per million per 
degree centigrade; negative. 

Methods of Mounting: Sandwich, air gap, clamped 
air gap, button. 

Advantages: The advantages and disadvantages 
of the BC cut are similar to those of the AC 
cut, except that the BC thickness-shear fre- 
quency constant is the highest obtainable for 
a rotated Y cut. A BC cut may thus have a 
greater thickness for the same frequency, and 
hence be less likely to be shattered from over- 
drive or mechanical shock — a distinct advan- 
tage at the higher fundamental frequencies 
where very thin crystals are used. Since the 
BC orientation is the negative angle of rota- 
tion which provides zero coupling between the 
shear modes, the element vibrates in a very 
pure length-thickness mode with an excellent 
frequency spectrum. For a given temperature, 
the electrical parameters of a BC crystal unit 
can be predetermined with an accuracy equal 
to or greater than that of the more commonly 
used BT units. 

Disadvantages: As in the AC cut, the principal 
disadvantage of a BC cut is its relatively large 


WADC TR 56-156 


39 



SMtion I 

Standard Quartz Elamants 

temperature coefficient. Because of this, the 
element is not widely used, and the zero-co- 
efficient BT cut, with an orientation suffi- 
ciently near to that of the BC cut to have a 
weak coupling between the shear modes, is 
used instead. An added disadvantage is that 
the magnitude of the rotation away from the 
Y axis is approximately double that for the 
AC cuts. For this reason the piezoelectric co- 
efficient is smaller for the BC than for the 
AC or AT cuts, and, hence, somewhat higher 
voltages are required to maintain oscillations. 


1-114. CHARACTERISTICS OF ELEMENT B 

Description of Element: BT cut; yxl: — 49°8'; 
length-thickness-shear mode ; or, yzw : — 49°8' ; 
length-width-shear mode. 

Frequency Range: 1000 to 20,000 kc (fundamental 
vibration) . 

16,000 to 75,000 kc (overtone modes). 
Frequency Equation: 


f 


ks/t (fundamental vibration when t « 1 and w) 


f = 


h* 



n‘ 

+ ai -p- -1- aj 


(P - D* 
w* 


where m, n, and p are integers. 

Frequency Constant: k, = 2560 kc-mm. 
Temperature Coefficient: 0.0 at 25°C; figure 1-50 


shows the total relative frequency deviation 
for the normal maximum, minimum, and 
average angles of this element; the temperar 
ture coefficient in parts per hundred per de- 
gree centigrade at each point on a curve is 
the slope at that point. Zero coefficients are 
obtained at 20°C and 76°C when ^ is — 49®16' 
and — 47°22', respectively. 

Methods of Mounting: Sandwich, air gap, clamped 
air gap, button. 

Advantages: The temperature-frequency charac- 
teristics make this element useful for high- 
frequency oscillator control where the tem- 
perature is not expected to vary too widely 
from the mean value. It is particularly ap- 
plicable for use in radio equipment which is 
to operate at the high end of the high-fre- 
quency spectrum. Most of the high-frequency 
crystal oscillators employ either the BT or the 
AT cut, with the B element, because of its 
larger frequency constant, often preferred at 
frequencies from 10 to 20,000 kc. Since the 
orientation angle is near that of the BC cut, 
the shear modes are not too strongly coupled 
together; and, when ground to proper dimen- 
sions, the B element exhibits a reasonably 
satisfactory frequency spectrum. 

Disadvantages: Like the A element, the B element 
is not suitable for use at the lower frequencies 
because of the large thickness dimensions 



- «0 40 ZO C to 40 *0 ao 4 100 

TEMPCKATURE (*C) 


WADC TR 56-156 


figure 1-50. Tumporuture-truguencY eharecturMcs of olemonf B 

40 




that would be requii'ed. Because of its grreater 
angle of rotation from the Y axis, the BT has 
a smaller piezoelectric coefficient than the AT 
cut, and hence requires a higher voltage to 
maintain oscillations. Also, the temperature 
coefficient of the BT cut increases more 
rapidly than that of the AT cut when the 
temperature varies to either side of the zero 
point. Moreover, zero temperature coefficients 
cannot be obtained at as widely separated 
temperature, as can be done with the AT cut 
by slightly varying the orientation angle. This 
limitation, however, becomes an advantage in- 
asmuch as it reduces the percentage error 
when cutting a crystal to provide a given tem- 
perature-frequency characteristic. The great- 
est problem in preparing a B element is to 
avoid those length and width dimensions that 
cause the frequencies of unwanted modes to 
approach the frequency of the desired mode. 
As in the case of the AT cut, a BT blank with 
a good frequency spectrum will require a cer- 
tain amount of trial and error in the finishing 
process. For the simpler modes of lower order, 
the following equations give the face dimen- 
sions of a BT cut which produce unwanted 
resonances of the same frequency, f, as that 
of the thickness-shear mode. 


For flexure modes along X: 

„ 1810 n , n A c \ 

X — ^ (n — 2, 4, 6, • . .) 

For shear modes along X : 

For shear modes along Z'l 

Z'= (n = 1, 3, 5, . . .) 


Section i 

Standard Quartz Elements 

With f expressed in kc, X and Z' are given 
in millimeters. {Z' is the dimension of the ro- 
tated Y cut that originally was parallel to the 
Z axis.) Either X or Z' may be the length, 
with the other dimension being the width. As 
in the case of the A element, a convex con- 
tour of a plate will aid in restricting unwanted 
modes. At 1000 kc the contour may be as 
great as 5 microns ; the thin, 20,000-kc plates, 
however, must be flat. The equations above 
hold only for flat plates, but are approxi- 
mately correct if the contour is very small. 

1-115. CHARACTERISTICS OF ELEMENT C 

Description of Element: CT cut; yxl or yzw: 37® 
to 38° ; face-shear mode. 

Frequency Range: 300 to 1100 kc. 

Frequency Equation: f = ki/w (fundamental of square 
plate). 



Frequency Constant: k, = 3070 kc-mm. (Square 
plates are preferred since they have fewer 
secondary frequencies.) 

Temperature Coefficient: 0.0 at 25° C for rotation 
angle of 37°40'. Figfure 1-51 shows the total 
relative frequency deviation with tempera- 
ture for maximum, minimum, and average 
angles of rotation for a nominal cut of 37°40'; 
the initial temperature is taken at 25°C. The 
slope of a curve at any point is the tempera- 
ture coefficient in parts per hundred per de- 
gree centigrade at that point. Note that as 
the rotation angle is increased, the zero co- 
efficient is shifted to a higher temperature; 



TEMPERATURE (*0 


WAOC TR 56-156 


figure 1-51. Temperature-frequmHcy characteristics of element C 

41 




Section I 

Standard Quartz Elements 

Z' 

the same is true when the ^ ratio is in- 
creased. For square plates, zero coefficients 
can be obtained at higher temperatures (50°C 
to 200°C) by rotation angles from 38‘’20' to 
41° 50', respectively. 

Methods of Mounting: Wire, cantilever clamp. 

Advantages: The CT cut is essentially a BT cut 
rotated approximately 90° so that the face 
shear of the C element corresponds to the 
thickness shear of the B element. This orien- 
tation provides a zero-temperature-coefficient 
shear mode for generating low frequencies, 
without requiring a crystal of large thickness 
dimension. The frequency characteristics of 
the C element, as compared with the D ele- 
ment, are roughly analogous to those of the 
B with the A element, except that the former 
pair vibrate at low frequencies, and the latter 
at high frequencies. The C element has the 
higher frequency constant, so is generally 
preferred over the D element at the high end 
of the low-frequency range. The C element is 
widely used both for low-frequency oscillator 
control and in filters, and does not require 
constant temperatures control under normal 
operating conditions. One of its principal ap- 
plications has been as the control element in 
frequency-modulated oscillators. 

Disadvantages: Because of its larger frequency 
constant, the C element must be cut with 
larger face dimensions than the D element to 
provide the same frequency of vibration. 
Thus, for the generation of very low frequen- 
cies the smaller DT cut is the more economical 
to use. Care must be taken that flexure modes 
are not strongly coupled to the face-shear 
mode. To prevent a coincidence of resonance 


between the two modes, the following thick- 
nesses have been used : 


Frequency Range in KC 
870 to 428 
428 to 476 
475 to 540 
730 to 876 
876 to 1040 


Thickness in Mils 
18.6 to 19.9 

16.0 to 17.6 
18.6 to 19.9 

12.0 to 14.0 

16.0 to 17.6 


1-116. CHARACTERISTICS OF ELEMENT D 


Description of Element: DT cut; yxl or yzw: 

—62° to —53° ; face-shear mode. 

Frequency Range: 60 to 600 kc. 

Frequency Equation: f = )^/'n (fundamental of square 
plate). 


f 




+ 


W* 


(m 

(n 


1. 2, 3, 

1 , 2 , 8 , 


) 

) 


Frequency Constant: ki = 2070 kc-mm. (Square 
plates are preferred since they, have fewer 
secondary frequencies.) 

Temperature Coefficient: 0.0 at 26°C for rotation 
angle of — 52°30'. Figure 1-62 shows the total 
relative frequency deviation with tempera- 
ture for maximum, minimum, and average 
angles of rotation for a nominal cut of 
— 62°30', where the initial temperature is 
taken at 26°C. The slope of a curve at any 
point is the temperature coefficient in parts 
per hundred per degree centigrade at that 
point. Note that as the rotation angle is in- 
creased, the zero coefficient is shifted to a 
higher temperature. The upper limit for a 
zero coefficient is approximately 200°C, when 
^ = -64°. 

Methods of Mounting: Wire, cantilever clamp. 



-«0 40 20 0 20 40 eo ao 4-IOO 

TEUPCSATURe CO 


WADC TR 56-156 


Figure 1-52. Tumpurature-fruquuiKy charaetuHsties of olomont 0 

42 




Section I 

Standard Quartz Elamontt 


Advantages: The DT cut is essentially an AT cut 
rotated approximately 90° so that the face 
shear of the D element corresponds to the 
thickness shear of the A element. This orien- 
tation provides a zero-temperature-coefficient 
shear mode for sreneratinsr low frequencies, 
without requiring a crystal of large thickness 
dimension. Because the frequency constant is 
less than that of element C, the face dimen- 
sions of element D are smaller for a given 
frequency, and hence the DT is the more eco- 
nomical cut for use at very low frequencies. 
Like the C element, the D element is widely 
used in both oscillators and filters, and does 
not require constant temperature control 
under normal operating conditions. 

Disadvantages: At frequencies above 500 kc, the 
impedance effects introduced by the mounting 
become excessive, since the contact surfaces 
between the crystal and the supporting wires 
become rather large compared with the area 
of the crystal faces. Hence, the C is preferred 
over the D element in the 500 — 1000 kc range, 
since the higher frequency constant of the 
former permits a larger crystal face. To avoid 
strong coupling of the face-shear mode with 
flexure modes, certain thicknesses must be 
avoided. For most frequencies, however, a 
thickness of approximately 17 mils is satis- 
factory. 

1-117. CHARACTERISTICS OF ET-CUT 
PLATES IN COMBINATION MODE 

w 

Description of Element: ET cut, with -y ratio ap- 
proximately equal to 1.0; yxl or yzw; 66°30'; 
combination of coupled modes with second 
flexural vibration appearing to dominate a 
face-shear harmonic. 

Frequency Range: 600 to 1800 kc. 

Frequency Equation: f = k/w (square plate). 

2k 

f = T-T- — (nearly square plate). 

I yv 

Frequency Constant: k = 5360 kc-mm. 

Temperature Coefficient: 0.0 at 75° C; see figure 
1-53 for total relative frequency deviation. 

Methods of Mounting: Wire; preferably mounted 
in vacuum. 

Advantages: Besides its zero temperature coeffi- 
cient, the principal advantage of the ET cut 
is its high frequen <7 constant, which is almost 
1.8 times that of the C element. This permits 
an effective extension of the frequency range 
for this type of plate and mounting. Optimum 
performance is obtained at temperatures near 



40 SO 60 70 SO 90 K>0 110 

TEMPERATURE IN DEGREES CENTI6RADE 

Figure 1-53. Temperature-fregueney tharaeterlsties 
of ET-cot plate vibrating in combination mode * 

75°C, so that the element is particularly ad- 
vantageous where crystal ovens are used. 

Disadvantages: Stability and general performance 
are inferior to those that can usually be ob- 
tained by using, according to the particular 
frequency, either an A or a C element. 

1-118. CHARACTERISTICS OF FT-CVT 
PLATES IN COMBINATION MODE 

Description of Element: FT cut, with w/1 ratio, 
approximately equal to 1.0; yxl or yzw; 
—57°: combination of coupled modes with 
second flexural vibration appearing to domi- 
nate a face-shear harmonic. 

Frequency Range: 150 to 1500 kc. 

Id 

Frequency Equation: f = — (square plate) ; 

w 

2k 

f = ; (nearly square plate). 

1 + w 

Frequency Constant: k = 4710 kc-mm. 

Temperature Coefficient: 0.0 at 76°C; see figure 
1-64 for total relative frequency deviation. 

Methods of Mounting: Wire ; preferably mounted 
in vacuum. 

Advantages: The advantages of the FT cut are 
approximately the same as that of the ET, 
except that the FT has a lower frequency 
constant. The FT is related to the ET in ap- 
proximately the same way that the DT is 


WADC TR 56-156 


43 




Section I * 

Standard Quartz Elements 



40 SO 60 70 eo 90 100 110 

TEMPERATURE IN DEGREES CENTIGRADE 

Figure 1-54. Temperature~trequency charaeterfstics 
of FT-cut plate vibrating In cembinrrflen mode * 

related to the CT. However, the frequency 
constants of the £T and FT are approxi- 
mately twice that of the low-frequency shear 
elements, so that these cuts can be made in 
practical sizes for twice the frequencies ob- 
tainable from the CT and DT crystals. Like 
the ET, the FT cut is particularly suited for 
use in ovens at temperatures between 70® 
and 80“C. 



temperature in degrees c 


Figure 1-55. Temperature-frequency characteri§tlc$ 
of element C '* 

Disadvantages: Stability and general performance 
are inferior to those that can usually be ob- 
tained by using, according to the particular 
frequency, either an A, C, or a D element. 

1-119. CHARACTERISTICS OF ELEMENT G 

w 

Description of Element: GT cut, with ratio -y = 

0.859; }7xlt: 61°7.6'/46°; width-extensional 
mode. 

Frequency Range: 100 to 660 kc. 

Frequency Equation: k,/w (fundamental). 

Frequency Constant: ki = 8870 kc-mm. 

Temperature Coefficient: Very nearly zero over 
the range from —26® to 4-76®. Figure 1-66 
shows the total relative frequency deviation 
from the initial frequency at 0®C. Note that 




■1 


■ 

■ 

■■ 

■1 

Kill 


■1 

■ 

■ 

■ 

■■ 

!■ 

II 


■r 

■ 

■ 

■ 

■ 

■ 

■■ 

■■ 

■■ 

1 

■ 


mi 

11 

m 

■ 

■ 

■ 

■ 

■ 

■ 

■ 

■ 

■ 

■■ 

■■ 

■■ 

■■ 

■■ 

■■ 

■ 

1 

■ 


rll 

■1 

■ 

■ 

■J 

■ 

■ 

■ 

■ 

■ 

■ 

■■ 

■■ 

■■ 

■■ 

II 

II 

■ 


-90 -60 -30 0 SO 60 90 

FIRST ROTATION ANGLE, IN DEGREES 


(B) 


Figure 1-55. (A) Diagram Illustrating the equivalence 
betvreen a face-shear mode and the length- and 
wldth-extenslonal modes of a rectangular plate which 
has been cut diagonally with respect to the face-shear 
element. (B) w/l ratio vs rotation angle, 5 , of element 
G providing sere temperature coefficient * 


WADC TR 56-156 




for a span of lOO^C the frequency does not 
vary more than one part in a million from 
the center frequency. The midpoint of the flat 
portion of the curve can be shifted from 25°C 
to SO’C by increasing the initial orientation 
angle from 61®7.6' to 61‘’30'; the zero coeflS- 
cient range will thus extend from 0°C to 
100°C. A temperature variation of ±15°C on 
either side of the midpoint will not change 
the frequency moreihan 0.1 part in a million. 

Methods of Mountinp; Wire, knife-edge clamp, 
pressure pins, cantilever clamp. 

Advantages: The GT cut provides tiie greatest 
frequency stability that has yet been obtained 
from a quartz plate. Where other quartz ele- 
ments have zero temperature coefficients at 
only one temperature, the G element will not 
vary more than one part in a million over a 
range of lOO’C. This element was originally 
suggested by the fact that a face-shear mode 
consists of two extensional modes coupled to- 
together. When a face-shear element is rotated 
45” the vibrations lose their shear effect and 
appear as two extensional modes — one along 
the width, and the other along the length. 
See figure 1-56 (A). Since all pure extensional 
modes must have a negative or zero tempera- 
ture coeflSdent, a positive coefiicient of a face- 
shear mode must be due to the coupling be- 
tween its two extensional components. If the 
cut of a face-shear crystal having a positive 
coefficient has been rotated 45”, the coupling 
between the extensional modes can be reduced 
by grinding down one edge so that one of the 
modes will increase in frequency. As the fre- 
quencies become more widely separated, the 
extensional modes will approach their true 


FABRICATION OF 

1-120. The production stages during the fabrica- 
tion of a crystal unit may differ somewhat from ' 
one manufacturer to the next because of variations 
in the instruments, techniques, and the type of 
mountihg employed. However, the general pro- 
cedure is fundamentally the same throughout the 
industry — ^first, the inspection and cutting of the 
raw quartz; next the lapping and etching of the 
diced blanks; and finally, the mounting and testing 
of the crystal unit in its holder. 

INITIAL INSnCTION OF RAW 9UARTZ 

1-121. The manufacture of a crystal unit begins 

WADC TR 56-1S« 


Sedien I 

Fabrication of Crystal Units 

negative temperature coefficients. At some 
ratio of width to length a zero coefficient will 
be obtained. The GT cut is properly a d:46” 
rotation of any positive-coefficient face-shear 
crystal in the Y group. Although the most 
satisfactory cut is the one described above, a 
number of other GT cuts have been investi- 
gated where the initial angle of rotation, 
has had negative as well as positive values. 
F.igure 1-66 (B) shows the w/1 ratios for both 
positive and negative angles that will provide 
a zero temperature coefficient. For negative 
values of the dominant mode is the one of 
lower frequency, whereas for positive angles 
of ^ the higher-frequency mode is dominant. 
The G element is used for the control of oscil- 
lators where the most precise accuracy is re- 
quired, such as in the frequency standards of 
loran navigational systems, the time stand- 
ards at the U. S. Bureau of Standards and at 
Greenwich Observatory, and in both fixed and 
portable frequency standards of general use. 
Other than in frequency and time standards, 
the GT cut is employed in filters that are de- 
signed for use under very exacting phase 
conditions. 

Disadvantages: The principal disadvantage of a 
GT cut is its expense. To obtain optimum 
temperature - frequency characteristics re- 
quires pains-taking labor in cutting and 
grinding to the exact orientation and dimen- 
sions. Furthermore, the excellence of a par- 
ticular cut will be of little advantage unless 
the mounting and the external circuit are also 
of superior design. For these reasons a G ele- 
ment is not particularly practical except when 
the utmost frequency precision is mandatory. 


CRYSTAL UNITS 

with the inspection of the raw quartz for impuri- 
ties, cracks, and inclusions. For this purpose, the 
arc lamp of the inspectoscope shown in figure 1-57 
is used. 

1-122. The inspectoscope tank is filled with a clear, 
colorless oil mixture having an index of refraction 
approximately the same as the average in quartz 
(1.52 to 1.56). Such a medium for transmitting the 
light to and from the raw crystal, or “stone,” is 
necessary in order to see the interior, for otherwise 
reflections and refractions at the rough surface 
will not only create an excessive glare, but will 
diminish the intensity of light penetrating beyond. 


4S 



..:pY|TAi^Nij fabrication 


Section I 

Fabrication of Crystal Units 





iw.A»Dfcsc?N.t' V 




eWTCH 


rtiCAftoEseewT, 
tAltP SWITCH U 


-".t; I 
f'S'^Vo.iiC 


/' j.t.tTivjtf*:', 






Figun 1-57. Po/ar/scope-/nspocfoscopo. Used tor oxaminlng raw quartz 


The lamp incorporates a high-powered (500- to 
1000- watt) projection system of white light, and 
inspection of the stone is performed by direct ob- 
servation. The usable parts of the stone are 
marked ; or if too many imperfections are present, 
the stone is discarded. 

INSPECTION FOR OPTIC AXIS AND 
OPTICAL TWINNING 

1-123. If a stone has retained some of its natural 
faces, the optic (Z) axis may be readily located by 
direct observation. In the usual case, however, it 
is necessary to use the plane-polarized light system 
that is provided by the inspectoscope. The light 
from a mercury or incandescent lamp is plane 
polarized by a Polaroid plate placed between the 
lamp and the tank. On the opposite side of the tank 
is a second Polaroid plate with its transmission 
axis perpendicular to that of the first, so that if a 
stone is not in the tank to rotate the polarity of the 
light, the rays will be stopped at the second plate. 
Light that does filter through, however, is re- 
flected upward by a mirror, and the pattern may 
be observed through the glass cover shown in 
figure 1-57. When a stone is placed in the tank and 
oriented so that its optic axis is parallel to the rays, 
the polarity of the rays will be rotated and a bright 

WADC TR 56-156 


image will be reflected from the mirror. If white 
light is used, a pattern of concentric rainbow colors 
will appear ; and if monochromatic light is used, a 
pattern of concentric rings of light and darkness 
will appear. The optic axis will be exactly parallel 
with the light rays when the stone is in the posi- 
tion that yields the fewest and broadest bands. If 
optical twinning is present, it will be revealed by a 
fine-toothed pattern cutting across the rings, as 
indicated in figure 1-58. The twinning areas are 
more clearly indicated when white light is used, 
and when viewed slightly off the optic axis. On the 
other hand, monochromatic light produces ring 
patterns of maximum clarity for the determina- 
tion of the optic axis itself. Flat surfaces are 
ground on opposite sides of the stone, parallel to 
the optic axis ; and, with the stone resting on one 
of the flat surfaces at the bottom of the inspecto- 
scope tank, a line is drawn on the upper surface to 
indicate the approximate Z-axis direction. 

USE OF CONOSCOPE FOR EXACT 

DETERMINATION OF OPTIC AXIS 

1-124. After the approximate optic (Z) axis is 
determined, the stone is cemented to a glass plate, 
and a small-end-section of the crystal is sliced off 
with a diamond saw, leaving a flat surface approxi- 


46 


/// nc 



' Section I 
Fabrication of Crystal Units 

mately perpendicular to the optic axis. The stone is 
then mounted on an adjustable orienting jig, 
which is placed against the reference edge in a 
conoscope tank. The conoscope (see figure 1-59) 
provides a polarized light system with which the 
optic axis may be accurately located by observing 
a concentric ring system. The principle of the 
conoscope is similar to that of the polarizing sys- 
tem of the inspectoscope, except that a converging 
lens system and a vernier system are provided that 
permit the optic axis to be determined with an 
accuracy of one degree. The handedness of the 
crystal is readily determined by rotating the 
second Polaroid plate, or analyzer, of the conoscope. 
The quartz is right or left according to whether 
the concentric rings appear to expand or contract 
for a given direction of analyzer rotation. When 
the Z axis is accurately determined, each end is 
trimmed to form plane surfaces (“windows”) 
exactly perpendicular to the Z direction. 

SECTIONING THE STONE 

1-126. There are three general methods of cutting 
the stone to obtain crystal blanks of desired 
orientation: the direct-wafering, X-block, and Z- 
section-Y-bar methods. In direct watering, shown 
in figure 1-60, wafers are sliced directly from the 
stone at the desired orientation, and the blanks are 
diced from the wafer. The X-block method, as in- 
dicated in figure 1-61, is similar to that of direct 

Figure t-58. Polarized-light view of pyramidai cap 

indicating optical twinning * 



figure T-59. Conoscope. Used for locating accurately the optic axis end for determining 

the handedness of quartz stones * 

WADC TR 56-156 47 , 



III stc. 




Sadion I 

PabricQtien of Cryctal Unita 



wafering, except that, before being sliced into 
oriented wafers, the stone is cut into one or more 
blocks with place surfaces at each end of the Z 
axis', and at the ends of one of the X axes; each 
surface is accurately cut at right angles to the axis 
it terminates. It is from these “X” blocks that the 
properly oriented wafers are cut and then diced 
into blanks. The third method of cutting proceeds 
as indicated in figure 1-62. The stone is sliced into 
Z sections (cross-sectional slabs with plane faces 
perpendicular to the Z axis) ; the Z sections are cut 
into Y bars (bars with the length parallel to the Y 
axis) ; and crystal blanks are sliced at the desired 

Figure 1-40. Dintt-wafaring method of cutting 
ciyataf Monk* 



MOTHER OUARTZ SECTIONED 
INTO Z BLOCKS 




RLANE OF PAPER) PLACED AGAINST 
REFERENCE EDGE OF MOUNTING JIG. 

WAFER CUT FROM X BLOCK 


FIgum 1-41. X-block method of cutting wafers from unfoced stone. Mfofers, on being diced, provide cryctal 

blankt at the proper orientation 


WAOC TR 56-156 


49 



Section I 

Fabrication of Cryctal Unit* 


2 SECTION 


LHQ MATCHING WINDOW 


LHQ MATCHING ARM 


RHQ MATCHING ARM 


RHQ MATCHING WINDOW 


marking template 


MARKING ARM 


QUARTZ 

SP^IMEN 


FILAMENT LAMP HOUSING 


NOTE. LHQ»LEFT-HfcND 0O*NTZ, RHO« RIGHT-HAND QUARTZ 


Figun 1-^3. Pinhole orlaseopo with mofehinfl ana marKing arms tor use on Z sections 
WADC TR 56-156 


Section I 

Fabrication of Crystal Units 

M 



Figure 1-64. X-ray deferminafion of X axis in Z 
block. M is horizontal bisector of angle that ray must 
make if reflected beam is to enter ionization chamber. 
6 , the Bragg angle of X-ray reflection for copper- 
anode Ka radiation, is predetermined according to the 
particular atomic plane to be identified. For plane 
that is parallel to an m face, and hence to an X axis, 
$ — 10° 38'. With positions of X-ray source and 
ionization chamber fixed, rotation of Z block about 
Z axis will cause maximum current to flow through 
ionization chamber when an X axis becomes 
perpendicular to M 



Figure 1-65. Reflection patterns of twinned Z section, 
showing both types of twinning. The section is pre- 
dominantly right quartz, but is fairly evenly divided 
by the electrical twins a and b. The small regions of 
optical twinning of one electrical sense are shown in 
C, and those of the opposite sense are shown in D. 
The X-axis polarities indicated apply only to the 
respective bright regions. The regions marked f 
contain flaws * 


figure 1-64, an X-ray beam is directed toward the 
crystal’s vertical surface, which deflects part of the 
beam into the window of an ionization chamber, 
causing a current to flow that has an amplitude 
proportional to the intensity of the rays entering 
the chamber. X-rays of constant wavelength are 
propagated in a narrow pencil from a properly 
filtered source, which consists of a special high- 
voltage cathode-ray tube having a copper anode. 
The X-rays are emitted by virtue of the high- 
energy electrons’ striking the copper target, and a 
thin nickel plate is inserted in the X-ray path to 
eliminate unwanted wavelengths. The atomic 
planes of the crystal lattice effectively serve as 
reflecting surfaces, except that interference be- 
tween the reflected rays from adjacent parallel 
planes eliminates all angles of reflection except 
those that permit the path lengths of coinciding 
rays to differ by an integral number of wave- 
lengths. The above condition is satisfied when the 
distance between the atomic planes is related to 
the wavelength and the angle of incidence of the 
X rays in a manner that can be expressed by 
Bragg’s law : 

nA = 2d sin 6 
where: n = 1, 2, 3, 

A = wavelength of X rays 

d = distance between parallel atomic 
planes 

e = angle of incidence of X rays with 
atomic plane 

’The ionization chamber is a gas-filled metallic 
cylinder having an electrode which is maintained 
at a voltage relative to the cylinder. X rays enter- 
ing the chamber will ionize the gas, permitting a 
current to flow through the external circuit. With 
6 predetermined for a particular atomic plane, the 
exact direction of the plane, and hence of the 
crystal’s orientation, can thus be determined by 
rotating the Z block for a maximum reading on 
the ammeter. 

CUTTING X BLOCK 

1-128. When the X direction has been precisely 
determined, the mounting jig is locked in position 
and transferred to a diamond saw, where windows 
are cut perpendicular to, and at each end of, the X 
axis — thereby forming an X block. After the align- 
ment of the windows is rechecked with the X-ray 
apparatus, the X block is cleaned, and then etched 
in 48% hydroflouric acid or a saturated solution of 
ammonium difluoride. 


WADC TR 56-156 


50 


DETERMINATION OF TWINNING 

1-129. Electrical and optical twinning boundaries 
can be observed directly by shining a spot lamp 
upon the etched Z windows of the X block. The 
light should be directed at approximately a 30- 
degree angle with the surface being examined, 
with the line of sight of the observer perpendicular 
to the surface. As the block is rotated about the 
axis perpendicular to the surface, there will be four 
particular orientations, each corresponding to a 
reflection of maximum brightness from the etched 
area of one of the four possible twins — right-hand 
quartz of either electrical sense, and left-hand 
quartz of either electrical sense. See figure 1-65. 
Normally a crystal is predominantly right or left, 
so that optical twinning usually appears only in 
small scattered regions. Electrical twinning, how- 
ever, normally divides a crystal into large regions 
of opposite electrical sense. The polarities of the 
various twinned areas can be readily determined 
by noting the angles of rotation at which maxi- 
mum brightness is observed. The axial polarities 
of an X block may also be determined by examin- 
ing the X windows with the aid of a pin-hole oria- 
scope having matching and marking arms designed 
especially for X sections. The images observed will 
differ according to the electrical sense of the 
particular area — also, according to whether hydro- 
fluoric acid or ammonium difluoride was used in 
etching. By a proper interpretation of the patterns. 


Sactien I 

Fabrication of Crystal Units 

the axial directions of the twinned regions can be 
suitably marked. If there is an excessive amount 
of scattered twinning, the block must be dis- 
carded ; otherwise, the observation permits a 
proper orientation for cutting slabs, so that opti- 
mum use of the quartz is possible. 

PREPARATION OF WAFERS 

1-130. The mounting jig, adjusted to the correct 
orientation, is transferred to a saw, and the X 
block is sliced into slabs of sufficient thickness for 
finishing. See figure 1-66. After being cleaned and 
etched, the slabs are inspected and marked for 
twinning, and the unusable portions are cut away 
by a diamond saw. Each slab is cemented to a 
holder and mounted in a jig for a final X-ray deter- 
mination of the orientation. The adjusted slab 
holders are transferred to the jig of a lapping 
machine, and the slabs are lapped on one surface, 
using an abrasive of 400-grain carborundum, until 
the lapped faces have the desired orientation. The 
slabs are then cemented to a large plate with the 
corrected faces down, and the uncorrected faces 
are lapped until parallel with the bottom faces. The 
“wafers,” as the slabs are now called, are next 
cemented to a glass-topped steel plate for dicing. 

PREPARATION OF CRYSTAL BLANKS 

1-131. The wafers are diced to the approximate 
crystal blank size with a dicing saw, as shown in 



Figur* 1-66. Diamond saw for cutting wafers from X block 


WADC TR 56-156 


51 



Saction I 

Fabrication of Crystal Units 




CRYSTAL (WRK 
HOLDER 
(C) 



figure 1-67. (A) Dicing saw. (B) Diced wafer. (C) Nest of lapping machine. (D) lapping machine 


figure 1-67. The dice are then transferred to the 
nest of a lapping machine, where they are lapped 
to a thickness equivalent to several kilocycles be- 
low the desired frequency. The lapping proceeds in 
three stages: rough, semifinishing, and finishing. 
However, the rough stage is accomplished prior 
to dicing, if the slabs are first lapped to wafers as 
described in paragraph 1-130. The semifinishing is 
done with 600-grain carborundum or equivalent, 
and the finishing requires 1000- to 1200-grain ab- 
rasive. Each of the last two stages should com- 
pletely remove the surface left by the preceding 
stage. (Where extreme care is required, as when 
finishing thin harmonic mode plates, 3000 mesh 
aluminum oxide mixed with water, cosmetic talc 
and powdered white rouge provides high-precision 
results, with the ultimate operating dependability 
greatly increased if the final lapping is followed by 



figure 1-68. Loaf of crystal dice, all blanks oriented 
in the same direction in preparation for edge grittding 
by edging machine * 


a brief semi-polishing with a mixture of water, 
rouge, and small amounts of rust preventative and 
wetting agent.) In the case of high-frequency 
blanks, the final lapping should bring the blanks 
within 25 to 50 kilocycles of the desired frequency. 
Next, a stack of 25 to 100 dice are clamped into a 
loaf, with all the blanks oriented in the same direc- 
tion. See figure 1-68. Two exposed edges are then 
ground parallel to locating surfaces by an edging 
machine. The loaf is then reversed and the two 
remaining edges are ground to square the blanks. 
Finally, the blanks are etched to the proper fre- 
quency. For high-frequency crystals, a frequency 
tolerance of ±0.002 percent will require that the 
finished blanks be etched to within ±0.00001 milli- 
meter of the ideal thickness. After cleaning, the 
crystal blanks are ready for mounting. 

METHODS OF MOUNTING CRYSTAL 
BLANKS IN CRYSTAL HOLDERS 

1-132. In the past, some confusion has resulted 
among radio engineers because of a mixed usage 
of the terms crystal holder and crystal unit by 
manufacturers in describing and naming their 
products. However, it is now generally agreed 
that the term crystal holder is to be used only in 
reference to the mounting and housing assembly, 
whereas the term crystal unit is to designate a 
complete assembly — that is, a crystal holder con- 
taining a mounted crystal plate. 

1-133. Crystal holders have been variously classi- 
fied by different specialists in the field, and in the 


WADC TR 56-156 


52 



absence of a standard nomenclature, a certain 
amount of overlapping has resulted among the dif- 
ferent classifications. The procedure to be followed 
here is to treat each particular method of mount- 
ing as a separate category. The holders to be dis- 
cussed are described according to the following 
types of mounting: gravity-sandwich, pressure- 
sandwich, gravity-air-gap, comer-clamped-air-gap, 
nodal-clamped-air-gap, dielectric-sandwich, plated- 
dielectric-sandwich, button-sandwich, knife-edge- 
clamp, pressure-pin, cantilever-clamp, solder-cone- 
wire, headed-wire, and edge-clamped. Only two 
general classifications of mounting, wire and pres- 
sure, are specified for Military Standard crystal 
units currently recommended for use in equip- 
ments of new design. The wire-mounted crystals 
are cemented directly to supporting wires. The 
pressure-mounted crystals are clamped in place 
by frictional contact with electrodes or other sup- 
porting elements. The wire mounts include the 
solder-cone-wire, the headed-wire, and the ce- 
mented-lead types, the latter being a particular 
kind of edge-clamped support cemented to the 
crystal. The pressure mounts include all other 
types listed above except the gravity-sandwich 
and the gravity-air-gap. 

Gravity Sandwich 

1-134. A “crystal sandwich” is simply a crystal 
plate sandwiched between two flat electrodes. In 
the simple gravity type of holder the crystal plate 
is placed on one electrode, with the second elec- 
trode resting on top and connecting to the external 
circuit through a flexible wire. See figure 1-69. 
A small clearance is provided around the sides to 
permit the crystal to vibrate freely. The clearance 
must not be too large, however, else the crystal 
will slide around in the holder, and may become 
chipped, or, at least, cause the electrical constants 
of the crystal unit to vary. The electrodes must be 
perfectly plane and made of noncorrosive metal, 
such as stainless steel, brass, or titanium. Brass is 
inferior to the other two metals, and titanium is 
largely a future possibility. The top electrode is 
considerably lighter than the bottom electrode, 
and is usually specifically dimensioned to fit a par- 


CRYSTAL^ 

• FLEXIBLE WIRE 

» 


1 

1 

4- \_ NON CORROSIVE 1 

METAL PLATES 


Flgun t-69. Gravity sandwkh 


SocHon I 

Fabrication of Crystal Units 

ticular crystal size. If the top electrode is too 
heavy, the impedance it introduces will be exces- 
sive, preventing the crystal from vibrating near its 
normal mode; on the other hand, if the top elec- 
trode is too light, firm contact will not be made 
with the crystal, and the operation of the crystal 
unit will be unstable. The edges of the crystal are 
slightly rounded to insure that they are free of 
burrs. Both the crystal and the electrodes must be 
perfectly clean, and the entire unit must be 
mounted in a hermetically sealed chamber. Nor- 
mally, the grid terminal of the unit connects to 
the flexible wire of the top electrode, and the 
ground or cathode terminal to the bottom electrode. 
1-135. The grravity holder was at one time widely 
used, but has now been largely replaced by holders 
that can maintain the crystal in a relatively fixed 
position if subjected to external vibration, such as 
might be encountered in vehicular or aircraft 
equipment. Occasionally, even when mounted in 
vibration-free equipment, a gravity crystal unit 
may fail to operate because one edge of the crystal 
has become closely pressed against one of the sides 
of the chamber. However, a light tap of the holder 
is usually sufficient to start oscillations. Compared 
with the holders in which flat inflexible electrodes 
are pressed against the crystal, the activity of the 
gravity holder is generally superior, and requires 
less voltage to maintain a state of oscillation. 

Prastnra Sandwich 

1-136. In holders of the pressure-sandwich type, 
the crystal is held more or less firmly between two 
flat electrodes under the pressure of a spring. In a 
typical assembly, the electrodes, which normally 
are of identical size and shape, are in turn sand- 
wiched between two very thin contact plates. The 
contact plates connect to two metal prongs that 
serve as electrical terminals and plugs for mount- 
ing the crystal holder in a socket. An insulating 
washer is placed over one of the contact plates, a 
coil spring is placed over the washer, and a neo- 
prene gasket is placed between the spring and the 
cover to provide hermetic sealing. Except for the 
glass spacers, the pressure holder described above 
is very similar to the air-gap holder shown in 
figure 1-70. 

1-137. Although the activity of a pressure-type 
crystal sandwich normally is not as great as that 
of the gravity type, it is much preferred because 
of its greater ruggedness and less critical require- 
ments concerning the orientation and vibration of 
the equipment in which it is to be mounted. 
Another advantage of the pressure holder is its 
relative simplicity of design — fewer of its compo- 


WAOC TR 56-156 


53 



Section I 

Fabrication of Crystal Units 




6LASS 

SPACER 

QUARTZ 
OSCILLATOR 
PLATE 




IIEOPRENE 



FLAT 

ELECTRODE 






(A) 




Figure 1-70. Methods, old and new, tor mounting crystal units. (A) Construction of early model crystal unit 
employing the gravity air-gap type of mounting, now largely outmoded. The crystal holder shown is the type 
FT-243. IBI Solder-cone wire mount for v-l-f len^h-ftexure crystal. (Courtesy HBEMCOI. (Cl Recently developed 
techniques for mounting shock-proof, 1-mc, A elements in the miniature HC-6/U metal holder to meet the 
specifications for 1-mc CR-IS/U crystal units: a. Reeves-Ho0man flexible nylon mount, b. Hupp loose-slotted 
edge-clamped mount, c. Bliley molded nylon bumper mount, d. RCA edge-clamping spring mount. (Courtesy 
McCoy Electronics.) 


WADC TR 56-156 


54 


nents require separate and exact dimensions for 
each particular frequency than is true of the ma- 
jority of holders. If the holder is constructed so 
that the sprinjr pressure may be adjusted, very 
slight variations in the frequency are possible ; the 
activity, however, will decrease proportionally as 
the pressure is increased. Although the pressure 
holder is widely used and is less expensive than 
most of the other types of holders, it has the dis- 
advantage of low activity and comparatively large 
damping of the oscillations. Thus, crystal units of 
this construction are not as sharply selective, nor 
as electrically predictable, as crystal units of more 
critical design. 

Gravity Air-Gap Mauating 

1-188. A gravity air-gap crystal unit is essentially 
a gravity sandwich, hut witti an air space separat- 
ing the crystal from the upper electrode. The air 
gap may be variable or fixed. In the variable type, 
the frequency can be adjusted slightly by raising 
or lowering the upper electrode, by means of a 
screw. The fixed air gap, however, is more com- 
monly used. As shown in figure 1-70, the fixed gap 
is maintained by glass or other insulating spacers 
placed between the electrodes. It is important that 
the thickness of the air gap not be an even quarter- 
wavelength of the acoustic vibrations which the 
crystal will generate in the air. Otherwise, the air 
waves, on reflection from the upper electrode, will 
return to the crystal 180 degrees out of phase with 
the normal vibrations, thereby introducing a high 
impedance and greatly reducing the activity. Maxi- 
mum activity is obtained when the air gap is an 
odd quarter-wavelength in thickness. The exact 
dimension, however, is not particularly critical in 
the case of shear modes, and a variation ot ±^i 
wavelength will cause little change in the ampli- 
tude of the vibrations. The quarter-wavelength 

formula is -r = -n, where v is the sound velocity 

in air, and f is in cycles per second. At room tem- 
perature and pressure, v = 330,000 mm/sec = 
12,992,000 mils/sec. The gap thickness should not 
exceed 3 mils, else the piezoelectric coupling will 
be too weak to maintain oscillations at reasonable 
voltages. Where it is necessary to have as large a 
piezoelectric coupling as possible, the air gap must 
be reduced to the minimum of ^ wavelength. 
1-139. The advantage of the air-gap mounting is 
that it shares the simplicity of design of the sand- 
wich holders, but eliminates the damping effect 
caused by the frictional contact of the upper elec- 
trode with the crystal. The presence of the air gap 
also effectively inserts in the crystal circuit a 


SocHen I 

Fabrication of Crystal Units 

series capacitance equal to that between the upper 
electrode and the crystal face, thereby increasing 
the ratio of the stored electrical to the stored me- 
chanical energy, and thus decreasing the electro- 
mechanical coupling. The reduction of the fric- 
tional losses (i.e., the effective electrical resistance) 
and the electromechanical coupling serves to give 
the crystal unit a higher Q, and thus to make it 
more selective and stable, and less affected by small 
variations on the external circuit. However, with 
the decrease in electromechanical coupling, the 
tendency of the crystal to vibrate is reduced ; and 
also reduced is the bandwidth for use in filters. 
On the credit side, the gravity-air-gap mounting 
is particularly convenient for the preliminary test- 
ing of crystals at the time they are being ground, 
and it has also been widely used in fixed-plant 
equipment with thickness-mode crystals operating 
at frequencies between 200 and 1500 kc. The prin- 
cipal disadvantages of the gravity-air-gap holder 
are the loose mounting of the crystal, the reduced 
piezoelectric coupling, and the possibility that a 
momentary overdrive will cause arcing across the 
air gap, or a corona discharge, thereby damaging 
the crystal and electrode surfaces, or even punc- 
turing the crystal completely. 

Coraer-Clampod, Air-Gap Mountiag 

1-140. The principal features of the corner- 
clamped, air-gap mounting are indicated in figure 
1-71. Note that air gaps exist at both the major 
faces of the crystal, except at the corners where 
the electrode risers, or. lands, clamp the crystal in 
position. If the length of a thickness-shear plate, 
such as an AT or BT cut, is not less than twenty 
times the thickness, a firm pressure may be ap- 
plied at the corners without greatly reducing the 
activity. The same precaution for avoiding an air 
column with dimensions approaching a multiple of 
a half-.wavelength is necessary for the clamped 
as for the undamped holder. The lands at the cor- 
ners normally provide a gap of 0.5 to 2 mils. 
1-141. The corner-clamped, air-gap method is 
widely used for mounting high-frequency thick- 
ness-shear crystals. Its operating characteristics 



Figure f-71. Typical ceriiar<lamp*d, air-gap mathod 
af maunling crystal* * 


WADC TR 56-156 


55 




S«c<ien I 

Fabrication of Cryatal Units 

are similar to those of the unclamped holder; and, 
in addition, it has the important advantage of 
clamping the crystal in a fixed position, thus per- 
mitting its use in aircraft and vehicular equip- 
ment. However, the clamping at the comers intro- 
duces an excessive amount of impedance when 
used for the lower-frequency, thickness-shear 
crystals where the 1/t ratio is less than 20 ; hence, 
this type of mounting is generally confined, to 
crystals with frequencies above 1500 kc. 

Nodal-Clamped, Air-Gap Mouatiag 

1-142. The principal features of the nodal-clamped, 
air-gap mounting are indicated in figure 1-72. ibis 
method may be used for mounting low-frequency 
piezoelectric elements vibrating in an extensional 
mode and having a nodal area at the center of the 
crystal. Each electrode has two risers for clamp- 
ing the crystal at each end of its nodal axis, and 
thus provides a secure mounting with a minimum 
of damping from direct contact between the crys- 
tal and the electrodes. A general advantage of any 
“zone-type” clamping, such as the nodal or comer 
methods where particular areas of a crystal are 
subjected to pressure, is that spurious frequencies 
requiring free vibrations in the clamped zones will 
be suppressed. 

Dielectric Saadwich 

1-143. This type of holder is essentially a crystal 
sandwich with a “lettuce” of high dielectric mate- 
rial inserted between the crystal and the elec- 
trodes. The sandwich and air-gap holders previ- 
ously described do not permit a crystal to operate 



FIgun 1-72. Typkal nodal-etampad, air-gap malhod 
of mouHfing ciystafs 



FIguro 1-73, Frossuro typo of dloloctrle sondw/cJi for 
mounting crystals 

near its elastic limit, for otherwise arcing would 
occur between the electrodes and the crystal. Low 
drive levels arc particularly necessary at frequen- 
cies above 4000 kc, for the likelihood of corona 
discharge or arcing increases with the frequency. 
Even if the arcing is insufficient to puncture the 
crystal, its presence will cause either wet or dry 
oxides to form on the crystal and the electrodes, 
thereby reducing the activity and greatly shorten- 
ing the crystal's useful lifetime, if not preventing 
its operation entirely. Many factors contribute to 
the possibility of a breakdown: type of holder, 
presence of sharp edges, smoothness and parallel-- 
ism of crystal and electrode faces, type of cut, air- 
gap dimensions, d-c and r-f voltages, frequency, 
and the like. However, since the arcing in all cases 
is the direct result of ionization of the air between 
the electrodes and the crystal, this danger may be 
removed if the more vulnerable air spaces are filled 
with an elastic cushion that has little tendency to 
ionize. It is necessary that the material have a 
dielectric constant much higher than that of air, 
and preferably higher than that of the crystal, 
and that it have low dielectric losses at the operat- 
ing frequency; otherwise the special advantages 
of the particular types of mounting with which 
dielectric material is used would be destroyed by 
an increase in damping. The dielectric filler may 
consist of insulating sheets cut to fit a particular 
mounting, or it may be coated over the electrode 
faces. In either case, a material of high dielectric 
constant will permit a crystal to be driven near 
its elastic limit without the danger of corona ef- 
fects, and with much less restraint of the normal 
vibrations than occurs when the crystal is in direct 
contact with the electrode faces. Suitable dielectric 
materials are mica, thin sheets of glass or fused 
quartz or other ceramics, “Cellophane,” nonsul- 


WADC TR 56-156 


56 



END VIEW SIDE VIEW 


Figur* 1-74. Centar-prassuiw typ* til dM»etric 
tandwich for mounting crystals 

furous rubber sheeting, cellulose esters and ethers, 
varnishes, lacquers, vitreous enamels, metallic 
oxides, rubber coatings applied by electro-deposi- 
tion, rubber containing resin and other fillings, or 
fused coatings of natural or synthetic resins. The 
sheets or coatings should be from 1 to 5 mils in 
thickness, but care should be taken that the thick- 
ness of the insulating material does not approach 
a multiple of a half-wavelength of the acoustic 
waves th^t will be generated. In any event, the 
addition of the dielectric material will tend to raise 
the impedance and frequency slightly, so that in 
extreme cases it may be necessary to grind the 
crystal to a frequency lower than that at which 
it is to operate. 

1-144. Figures 1-78 to 1-77 indicate different 
methods in which the dielectric fillers may be used 
in mounting a crystal. Figure 1-73 illustrates a 
pressure tjTpe of mounting with two sheets of di- 
electric material — arnica, for instance — inserted be- 
tween each electrode and crystal. Note that the 
mica extends beyond the edges of both electrodes. 
This feature is important, for although in a well- 
designed pressure sandwich no air spaces exist 
between the crystal and electrode faces, so that 
ionization and arcing do not occur at the major 
surfaces, corona discharges can and do occur at 
the edges, particularly if the sides of the chamber 
are close in, as is usual, causing the alternating 
field around the edges to be more intense. How- 
ever, with the insulating sheet of high dielectric 
constant overlapping the electrode edges, the in- 
tensity of the electric field will be greatly dimin- 
ished. That part of the dielectric directly between 
the crystal and the relatively inelastic electrode, 
acts as an elastic cushion, and thus serves much 
the same function as an air gap, but without in- 
creasing the possibility of corona or arcing effects. 
1-145. Figure 1-74 shows a top and a side view of 
a center-pressure typo of mounting, where two 
circular electrodes of small cross section are sepa- 
rated from the crystal faces by insulating sheets 


Section I 

FabricaHen of Crystal Units 

of high dielectric constant. Again, it is important 
that the insulation extend well beyond the edges 
of the electrodes. This arrangement increases the 
length of the shortest possible arcing path, and, in 
so doing, diminishes the chance of the occurrence 
of a discharge. 

1-146. Figure 1-75 illustrates two methods by 
which a comer-clamped air-gap holder can be con- 
verted into a dielectric sandwich while still retain- 
ing the principal advantages of the air-gap mount- 
ing. Figure 1-75A shows an insulating sheet cut 
to the dimensions of the air gap, and figure 1-76B 
shows a corner view of the assembled sandwich. 
Figure 1-75C is a side view of a similar sandwich. 




(C) 


FIguro 1-75. Two mothodt (B and C) by which a 
tornor-elampod, air-gap holdof can bo convortod into 
a diolottrle tandwieh. FIguro A ahowt a dioloctrle 
thoot cut to tho dimensions of tho air gap 


WAOC TR 56-156 


57 



Section I 

Fabrication of Crystal Units 



Flgun 1-76. Oieioctric sboot cut to Ml air gap at 
nodal-elampad mounHag 

but with two additional insulating sheets inserted 
to cushion the crystal entirely from direct contact 
with the electrode risers. 

1-147. Figure 1-76 illustrates the cut of an insu- 
lating sheet for converting a nodal-clamped, air- 
gap mounting into a dielectric sandwich. Two 
niches in the edges of the sheet are cut to fit the 
two risers of an electrode. When assembled, the 
sandwich is similar to the comer-clamped model 
of figure 1-75B ; or, if additional rectangular sheets 
are inserted next to the crystal, the assembly will 
resemble the arrangement in 1-76C. With either 
method, maximum rigidity is obtained for the 
nodal mounting with a minimum in damping. 

1-148. Figure 1-77 is a cross-sectional view of a 
gravity type air-gap mounting with the electrodes 
coated with an insulating material of high dielec- 
tric constant. It is characteristic of air-gap holders 
that the smaller the thickness of the air gap, the 
higher the r-f voltage that can be applied before 
arcing occurs between the crystal and electrode 
faced. When the electrodes make perfect contact 
with the crystal, not only are the opposing sur- 
faces theoretically at the same potential, but no 
ionizable substance lies between them in which an 
arc can form. However, the introduction of an air 
gap not only inherently reduces the electrome- 
chanical coupling of a crystal unit, but also effec- 
tively lowers the voltage that can be practicably 
applied. To remove the latter restriction without 
diminishing the advantages the air gap provides, 
the arrangement shown in figure 1-77 can be used. 
Note that the coating covers the edges of the elec- 
trodes — an important consideration since it is at 
the points of sharpest curvature that ionization 
is most likely to arise. 

1-149. The use of insulating sheets and coatings 
of high dielectric constant permits a crystal to be 
operated near its elastic limit without the danger 

WADC TR 56-156 59 



figuia 1-77. Cross-socf/onol vi»w of gravltf-air-gap 
mounfing with olactrodo surfacos protottad by coating 
of high-dioloctrk matorial 

of arcing, and hence this type of crystal unit can 
be operated at higher drive levels than would 
otherwise be possible. The dielectric sandwich 
would be advantageous in filter circuits where high 
amplitude signals are to be encountered; or in 
small portable transmitters where several ampli- 
fier stages are not possible, and the excitation level 
must be as high as possible ; or in any radio trans- 
mitter designed to be keyed in the oscillator stage 
where it is important that the oscillations built to 
peak amplitude in a minimum number of cycles. 
The insertion of the dielectric sheets also improves 
the stability and selectivity of the sandwich-type 
holders, inasmuch as they eliminate direct contact 
between the crystal and the relatively inelastic 
electrodes. The principal disadvantages of the di- 
electric sandwich are the reduced piezoelectric 
coupling caused by the separation of the electrodes 
from the crystal, and the damping effect of the 
frictional and small dielectric losses which are 
slightly greater than those of the air — provided 
the crystal is operated well below its elastic limit. 

notod-DloIcctric Sandwich 

1-160. This type of mounting is essentially the 
same as the previously described dielectric sand- 
wich except that a thin layer of conducting mate- 
rial is interposed between the dielectric sheets and 
the electrodes, or between the dielectric sheets and 
the crystal, or both. The conductive surfaces may 
be strips of metal foil not more than 1 mil in thick- 
ness, or they may be plated, painted, or sprayed 
directly on the insulating material. Suitable con- 
ducting substances are copper, nickel, silver, gold, 
platinum, and their alloys. The conducting layer 
may be coated on one or both major surfaces of the 
insulating sheet, or it may completely cover the 
edges as well as the major surfaces, thus effee-. 
tively converting the sheet into a highly compliant 
metal plate. 

1-151. Figure 1-78A illustrates the corner-clamped- 
air-gap mounting using dielectric plates having 
conducting films on both major surfaces. The two 


t 




ftgun 1-7B. (A) Air-gap mounting u§ing dMattrk 
shoots having conducting Alms on ooch major surloco. 
(B) Air-gap mounting using dlo/octrlc shoots having a 
tondueting ft/m on major suifoco In contact with 
o/octrodo. /Cl Dioloetrk sandwich In which loavos 
of motal foil aro Insortod botwoon dioloetrle 
and Cfystol 

films that are in direct contact with the electrode 
risers prevent the establishment of differences of 
potential across the air gaps, and hence obviate the 
possibility of arcing or corona discharges in these 
spaces. Figure 1-78B illustrates the nodal-clamped, 
air-gap mounting using dielectric plates having a 
conducting film on only one surface. In the case 
of air-gap mountings, if only one conducting sur- 
face is to be interposed between a crystal and each 
electrode, it is preferable that this surface make 
contact with the electrode rather than the crystal, 
so that possible electric fields will be “shorted” 
around the air gap. Figure 1-78C illustrates a di- 
electric sandwich mounting in which leaves of 
metal foil are inserted between the dielectric plates 
and the crystal. The metal foil, being very thin 
and flexible, snugly fits the crystal surface and in- 
terferes but little with the crystal’s vibrations. On 
the other hand, its presence insures a uniform po- 
tential at all points on the crystal's surface, thus 
protecting the surface from the effects of exces- 
sive electric stresses. 

1-152. The plated-dielectric sandwich combines 
the advantages of the plain dielectric sandwich 
with improvements in the frequency stability, 
crystal life, frequency spectrum, and piezoelectric 
coupling. The improvement in frequency stability 
is greatest in the case of the air-gap crystal units, 
for the danger of arcing or corona discharge in an 
air gap is removed without the insertion of a di- 
electric sheet to fill the gap. Since the damping 
effect of the air is less than that of the insulating 
material, the use of conducting film permits a 


Soctien I 

/ Fabrication of Crystal Units 

closer approach to the high selectivity of the pure 
air-gap mounting for crystals which are to be 
driven near their elastic limit. 'The insertion of a 
metallic film next to the crystal surface serves to 
reduce possible electrical stresses at the surface 
which might indirectly aid the production of small 
fractures, or cause ionization and chemical effects 
that would lead to a weathering of the crystal’s 
face. The insurance of a uniform potential at all 
points on the surface of the crystal also improves 
the frequency spectrum, particularly at very high 
frequencies, where many possible overtone modes 
can vibrate at frequencies close to that of the de- 
sired mode. However, the majority of the unwanted 
modes will have changes of phase and differences 
in amplitude along the major plane of the crystal, 
so that the resulting eddy currents that they in- 
duce in the conducting surfaces will aid in damp- 
ing them out. Where the interfering modes might 
otherwise lead to a frequency drift or jump, the 
damping effect will be reflected principally as an 
increase in the effective electrical resistance and 
as a decrease in activity. Finally, closer piezoelec- 
tric coupling is achieved if the entire insulating 
material is given a metallic coating. The dielectric 
sheet thus effectively becomes an extension of the 
electrodes, and the close coupling of the simple 
sandwich mounting is approached, but without the 
heavy damping caused by friction between the 
crystal and solid metal. 

Battan Manats 

1-153. The ceramic button crystal mount repre- 
sents the ultimate in crystal-holder design yet to 
be reached via the sandwich and air-gap evolu- 
tionary chain. Originally, the button electrode was 
developed as an all-metal modification of the cor- 
ner-clamped, air-gap type. As illustrated in figure 
1-79A, the all-met^ electrode is provided with 
conventional lands at the corners, but the effective 
center area has been reduced by surrounding the 
center with a relatively deep circular groove. The 
effect is to reduce the shunt capacitance across the 
crystal while retaining a central area of sufficient 
size for adequate excitation; the reduction in 
shunt capacitance is particularly desirable if the 
crystal is to be operated in the v-h-f range. Also, 
since the principal excitation is confined to a cen- 
tral circular area, the likelihood of spurious modes 
is somewhat reduced, because the vibrating part 
of the crystal tends to exhibit the properties of a 
circular plate. The superior frequency spectrum of 
the circular plate is probably even more closely ap- 
proached by using electrodes having solid ring- 
shaped lands that completely surround the circular 


WAOC TR 56-156 


59 



Section i 

Fabrication of Crystal Unite 


(A) 


(B) 



ALL-METAL 
BUTTON ELECTRODE 

(C) 

I 

CRYSTAL HOLDER 
HC-IO/U 


ASSEMBLED UNtT 




ALL-METAL 
SOLID -LAND 
ELECTRODE 



NICKEL- SILVER 
BARREL 



SPRING ASSEMBLY 
AND CONTACT- MOUNTED 
HERMETIC SEAL 



CONTACT- MOUNTED 
HERMETIC SEAL 


CERAMIC -BUTTON ELECTRODES 


FIgun 1-79. Button electrodos and mmthodt 
of mounting 

air gap at the center. See figure 1-79B. However, 
it is by combining the advantages of the button 
mounting with those of the plated-dielectric sand- 
wich and circular quartz plates that optimum per- 
formance is obtained for thickness-shear modes at 
very high frequencies. Figure 1-79C shows the 
principal parts and the complete assembly of Crys- 
tal Holder HC-IO/U. The shunt capacitance is held 
to a minimum, first, by the use of ceramic sup- 
porting plates in place of all-metal electrodes, and 
second, by the use of a coaxial electrode system 
in place of the conventional method where the 
crystal leads parallel each other through the base 
assembly. The ceramic-button electrodes are usu- 
ally very thin metallic platings that cover a small 
circular area at the center of each ceramic plate. 
Although the lands may be provided by forming 
thickened sections at the rim of the ceramic disks, 
usually they are obtained by plating metal risers 
on the surface of the ceramic; these plated risers 
are not connected electrically to the center metallic 
section. The air-gap thickness is normally between 
three and five microns. A notch in each ceramic 
button permits an extension of the electrode plat- 
ing to the opposite side, so that contact with the 
crystal leads can be made with a minimum of in- 
crease in electrode capacitance. This type of crys- 


tal holder is unequalled in performance when used 
with harmonic-mode crystals in the very-high-fre- 
quency range. It should be noted, however, that 
one of the original advantages of the plated-dielec- 
tric sandwich mounting is not fully realized in the 
case of ceramic-button electrodes — namely, the 
protection against arcing or corona discharges. For 
this reason, the ceramic-button crystal units will 
not withstand as high a drive level as might other- 
wise be possible. On the other hand, the thin air 
gap that can be obtained, the presence of a high- 
dielectric material almost flush with the edges of 
the plated electrodes, and the firm mechanical sup- 
port by which the crystal is held and cushioned 
against shock make this unit more durable under 
high drive levels than conventional air-gap holders. 
One of the more important advantages of the 
ceramic-button is the reduction of spurious modes 
through the use of circular quartz plates and small 
electrodes. The small electrode dimensions serve to 
concentrate the activity at the center of the crys- 
tal, where the crystal is most likely to be of uni- 
form thickness; thus, sudden frequency jumps are 
prevented, for these seem to be due primarily to 
abrupt shifting of the center of activity between 
areas having slightly different average thicknesses. 

Plated Crystals 

1-154. Since 1940 the designers of crystal units 
have increasingly favored the use of electrodes in 
the form of extremely thin metal films deposited 
directly on the crystal. Coatings of silver and gold 
have been successfully applied by spraying and 
baking, but in general the most advantageous 
method is by evaporating the metal in a vacuum 
and allowing it to condense on the exposed sur- 
faces of the crystal. Sputtering processes are being 
used increasingly, particularly for base plating, 
where the crystal is plated in vacuum by ionic 
bombardment from high-voltage negative elec- 
trodes composed of the desired plating metal. Elec- 
troplating of crystals also finds application. The 
noble metals, gold and silver, are the elements 
most commonly used in plating crystals because of 
their resistance to oxidation, their relative ease of 
plating, and the strength of their soldered junc- 
tions. Other metals that are used in plating are 
nickel and copper. Aluminum plating is preferred 
if a crystal is to be held in position by pressure 
pins or knife-edge clamps. This is because alumi- 
num is more durable to frictional wear, and be- 
cause its lesser density permits an electrode of 
lighter weight. However, alumimun is the more 
difficult to apply, has a tendency to oxidize, and its 
soldered connections are not as strong as those of 


WADC TR 56-156 


60 





( 

silver or gold. For these reasons, silver is more 
widely used if the crystal is to be soldered between 
wire supports, and gold is used if the wire-sup- 
ported unit requires maximum stability and 
resistance to aging. Aluminum coatings are com- 
monly applied at 1 milligram per square inch, 
which is equivalent to a thickness of approxi- 
mately 0.0225 mil; silver is applied at 4 milligrams 
per square inch, a thickness of approximately 
0.0232 mil ; and gold is applied at 3 milligrams per 
square inch, a thickness of approximately 0.0114 
mil. The actual plating procedure may be divided 
into two or more steps involving more than one 
plating process. As an example, the Signal Corps 
Engineering Report E-1108 by J. M. Roman rec- 
ommends as many as three different plating stages 
during the fabrication of low-resistance, 50-mc 
harmonic-mode crystal units of the CR-23/U type. 
The base plating is accomplished by a sputtering 
machine in which a group of crystal blanks are 
mounted in a rectangular metallic mask midway 
between two gold electrodes, which are 51/2 inches 
square and inches apart. A bell jar is placed 
over the electrode assembly and is evacuated to 
0.05 to 0.02 millimeters of mercury. 2200 volts dc 
are applied between the crystal mask and the elec- 
trodes; first for 30 minutes at 100 ma with the 
mask negative in order to clean the crystals by 
ionic bombardment, and next, for 37 minutes at 
100 ma with the electrodes negative for the actual 
gold plating operation. A second sputtering ma- 
chine is used to clean the crystal mask of the gold 
deposited upon it during the plating procedure. 
This latter operation requires an hour at 100 ma. 
After being mechanically mounted on HC-6/U 
bases between 9-mil, edge-clamping spring wires, 
the crystals are given a preliminary performance 
test. If a crystal is more than 0.1 per cent off its 
nominal frequency it is subjected to an additional 
plating process, "niis time it is plated electrolyti- 
cally with nickel. The electroplating of 50-mc 
crystals proceeds at a rate of 0.9 ma, which is 
equivalent to a harmonic frequency change of 100 
kc per minute. The electrolytic solution consists of 
chemically pure nickel ammonium sulphate, boric 
acid, ammonium chloride, and water in a weight 
ratio, respectively, of 75/16/15/1000. After mount- 
ing, testing, and electrically bonding the plated 
crystal electrodes to the supporting wires with sil- 
ver cement, the crystals are given a final spot 
plating with gold in an evaporation type plater to 
bring them to their specified frequency. This final 
plating process is accomplished in vacuum while 
the crystal is connected in a test oscillator circuit. 
1-155. The advantages of using metal-film elec- 


SecHen I 

Fabrication of Crystal Units 

trodes are several fold: maximum piezoelectric 
coupling is achieved ; the possibility of arcing be- 
tween the electrodes and the crystal is reduced to 
a minimum ; variations of frequency due to a shift 
of the position of the crystal relative to the elec- 
trodes are eliminated; frictional losses and wear 
due to inelastic contact between the crystal and the 
electrodes are removed; the metallic film aids in 
protecting the crystal from erosion; the film is 
readily adaptable for various types of nodal mount- 
ing, and is easily divided into several electrodes 
for use in exciting particular harmonic modes. 
All in all, the plated crystal is the most practical 
for obtaining optimum crystal performance at 
low and fundamental-mode high frequencies. The 
metal-film electrodes, however, have certain dis- 
advantages; the metal has a tendency to absorb 
moisture, thereby causing the frequency to 
change ; when clamp supports are used, friction at 
the clamped points will eventually wear away the 
metal coating; and generally, the mounting tech- 
niques are more critical for plated crystals. 

Pressure-Pin Mounting 

1-156. Pressure-pin holders (see figure 1-80) are 
used to support low-frequency (up to 200 kc), 
electrode-plated crystals, particularly those crys- 
tals used in telephone filters. Each crystal is 
clamped at the center of a nodal zone by one or 
more pairs of opposing pins. For crystal plates 
one-half inch square and smaller, the diameter of 
the pins is approximately 10 mils, and the clamp- 
ing force varies from one to two pounds ; for larger 
plates, pins of larger dimensions exerting some- 
what greater clamping forces are used. It is im- 
portant that the plated electrode be of aluminum, 
for the greater hardness of aluminum is required 
to resist the wear at the points of contact with the 
pins. Normally, these holders are designed for 
mounting a complete set of filter crystals within 



Figun 1-80. Pressur 0 -pin mounting, with provltioni 
for supporting four platod filtor trystals * 


WADC TR 56-156 


61 


Section I 

Fabrication of Crystal Units 


a single hermetically sealed container. The holder 
shown in figure 1-80 mounts four crystal elements. 
'I’he pressure is applied by the springs mounted at 
the ends, and the pins serve as electrical connec- 
tions as well as mechanical supports. For greater 
mechanical stability, slight niches may be made in 
the quartz at the clamped points. 

1-157. The pressure-pin holder has the advantages 
and disadvantages of the plated electrodes, and is 
used primarily for low- and medium-frequency 
filter crystals. It is particularly applicable for use 
with face-shear elements, since these have but one 
practicable nodal spot for clamping. The chief 
limitation of the pressure-pin mounting is the 
mechanical impedance it introduces. If the diam- 
eter of the pin is made too small, the crystal will 
tend to rotate about its axis of support; however, 
the larger the diameter is made, the more the 
clamping area will extend beyond the nodal point. 
To obtain optimum performance with this type 
of mechanical support, a resonant-cantilever de- 
sign foi’ pressure pins was invented by J. M. 
Wolfskin of the Bliley Elect’dc Company (U.S. 
Patent 2,240,453, 1941). This step was quite sig- 
nificant, not only in its own right, but because it 
provided a forerunner of the resonant-wire type 
of mounting. The following discussion is based 
on an analysis of the cantilever clamp by R. A. 
Sykes (Bibliography No. 741). 

The Cantilever Clamp 

1-158. The cantilever clamp is a pressure-pin sup- 
port designed to resonate at or near the crystal 
frequency. Figure 1-81 illustrates a pin mounted 
as a cantilever, and figure 1-82 indicates the 
motion of the cantilever as a quarter-wavelength 
bar with a node at the fixed base and a loop at the 
point of contact with the crystal. It is important 
that even quarter-w'avelengths be avoided, else the 
mechanical energy returning to the crystal will 



Figure 1-81. Cantilever clamp for providing a 
resonant-pin support for the crystal * 



Figure 1-82, Resonant motion of cantilever pin when 
its length is equal to one-quarter wave-length of 
clamp-free flexural vibration. Note that the effective 
free end of the pin is that end supporting the crystal 
(not shown) * 

oppose its motion, thereby greatly increasing the 
impedance and lowering the activity. The length of 
a cantilever pin that will present a loop to the 
crystal can be determined approximately from the 
frequency formula of a clamp-free rod in flexural 
vibrations : 

, mMv 

“ 8ir 1- 

where: m = 1.875 for the 1st node of vibration of 
the rod (pin) 

m = ^n — for n = 2, 3, . . . 

d = diameter of pin 
V = velocity of propagation along pin 
1 = length of pin 

For phosphor-bronze pins, v = 3.6 x 10* cm/sec; 
therefore, to support a 100-kc crystal, pins 1 mm 
in diameter should be 2.25 mm long to resonate in 
the mode indicated in figure 1-82. To resonate as a 
three-quarter-wavelength rod, n = 2, and 1 = 5.67 
mm for a pin of 1-mm diameter. The pin should be 
rounded at the end, as shown in figure 1-81, so that 
firm contact is made without the risk of having all 
the clamping force concentrated momentarily at a 
sharp point. 

1-159. A properly designed cantilever clamp should 
extend the useful range of the pressure-pin type of 
mounting to somewhat higher frequencies, and 
this has proved to be true in actual practice; how- 
ever, at the present time no data is available con- 
cerning its frequency application above 350 kc. 
Theoretically, a pair of pins could be used at any of 
their clamp-free harmonic modes, and thus pins of 
the same design need not be restricted to use at a 
single frequency. The principal promise of the 
cantilever clamp, however, is that it can provide a 
firm mechanical support while presenting a mini- 
mum of interference to the normal vibration of the 
crystal. 

Knife-Edged Clamp 

1-160. The knife-edged clamp is similar to the 
pressure-pin method of mounting, except that the 


WADC TR 56-156 


62 



Section I 

Fabrication of Crystal Units 


QUARTZ CRYSTAL PHOSPHOR BRONZE 

LENGTH i TO PAPER SPRING 



Figure 1-83. Knife-edge clamp support for two 
crystal plates. Each crystal has two pairs of plated 
electrodes, and is so mounted that each pressure 
blade makes electrical contact with a different elec- 
trode. This arrangement effectively provides four 
crystal elements for use in a balanced filler circuit * 

clamping prongs are blade-shaped, as indicated in 
figure 1-83. The dimensions of the clamping points 
are, on the average, about 35 mils in length, and 10 
to 15 mils in width. These blades are used with 
crystal elements that have a nodal axis parallel to 
the plane of the major faces, and care must be 
taken to make certain that the blades are centered 
along the nodal line. Pressure is applied by phos- 
phor-bronze springs, with the blades serving as 
electrical connections as well as mechanical sup- 
ports for the crystal. The holder shown in figure 
1-83 mounts two crystals, but, because the plated 
metal films are divided to provide two electrode 
pairs for each crystal, the equivalent of four crys- 
tal plates is effectively available for use in a 
balanced filter circuit. 

1-161. The advantages of the knife-edge clamp are 
essentially the same as those of the pressure-pin 
mounting, except that the greater surface of con- 
tact between the crystal and the clamp permits a 
firmer mechanical support. However, the knife- 
edge clamp is limited to use with those crystal 
elements that have W’ell-defined nodal lines. Its 
most important application has been as a mounting 
for the —18"' X-cut filter crystal, a crystal that 
can vibrate in a very pure length-extensional mode, 
and which has a nodal axis at the center parallel to 
the width dimension. The knife-edge clamp is gen- 
erally useful only at frequencies below 120 kc. 

Wire Mounting 

1-162. Wire-mounted crystal units are of two 
kinds: those that employ wire supports designed 
to resonate at the crystal frequency in a manner 


similar to that described in paragraph 1-158 for 
cantilever clamps, and those that clamp the crystal 
at the edges by non-resonant spring wire. This 
latter type of wire support is the cemenfed-lead 
mount, which is classified here as an edgevclamped 
mount. The wire mounting provides a firm but 
flexible support that serves to cushion the crystal 
from external vibration and shock. In addition, it 
can combine the advantages of the metal-film elec- 
trode with the low impedance of resonant supports, 
and can be used to mount any of the crystal 
elements, both high- and low-frequency plates, 
vibrating in extensional, shear, or flexural modes. 
Because of these advantages, the wire-type mount- 
ing is generally favored for crystal units used in 
military equipment. 

1-163. There are two principal types of resonant 
wire mounts, the solder-cone and the headed-wire. 
In general, the solder-cone support is restricted to 
relatively small crystal plates — for example, to 
frequencies above 300 kc for C elements. The 
headed-wire type is more suited to larger plates. 

Solder-Cone Wire Support 

1-164. A diagram of the solder-cone type of wire 
mounting is shown in figure 1-84, and a mounted 
crystal is shown in figure 1-85. The crystal to be 
mounted is first spotted with small silver footings, 
40 to 90 mils in diameter, at the nodal points 
where the wires are to be attached. Next, the elec- 
trodes are plated on the crystal by an evaporation 
or other process. Silver is generally used, although 
gold may be preferred where i-esistance to corro- 
sion is paramount. Aluminum has not been widely 
used in wire-mounted units, because of the weak 
junction it makes with the solder. However, recent 
experiments indicate that an aluminum junction 
with a solder of indium (a rare, fusible metal, 
chemically similar to aluminum) is quite strong, 
so that eventually greater application may be 
found for aluminum-plated crystals. The mount- 
ing wires are normally of phosphor bronze, be- 
cause of its high tensile strength and resistance 
to fatigue. A eutectic tin-lead solder is used that 
would normally be an alloy of approximately 63 
percent tin and 37 percent lead by weight; how- 
ever, to prevent an excessive diffusion of silver 
molecules from the silver spot into the solder dur- 
ing the soldering operation, the solder should con- 
tain 0.1 percent silver if the soldering is performed 
by hot-air blast, or a 59.5 — 34.5 — 6 percent tin- 
lead-silver combination if performed by hot iron. 
A solder cone in the shape of a bell (see figure 
1-84) has been found to provide the best perform- 
ance characteristics, and is the type of cone that 


WADC TR 56-156 


63 




Section I 

Fabrication of Crystal Units 



Figure 1-84. Solder-cone resonanf-wire support. The solder ball "tunes" the wire to the crystal frequency if 
it is placed at a distance equal to an odd multiple of a quarter wavelength (I,, 1^, etc.) 
from the peak of the solder cone * 


is least likely to rupture at the peak to form a 
"crater.” For small crystals, the part of the wire 
enclosed by the cone may be straight, but for 
larger crystals sufficient anchorage requires that 
the end of the wire form a small hook. The wire 
is tuned to resonance by fixing the position of a 
solder ball at an odd quarter-wavelength from the 
peak of the cone; the solder ball serves as a 
"clamped” point for reflecting the wave energy 
back to the crystal. The "free” end of the wire is 
effectively at the point where it enters the solder 
cone. The distances h, I™, and I, indicated in figure 
1-84 mark "free lengths” of wire that will be reso- 
nant at the given wavelength. Note that each of 
the lengths defines a distance from the "free” end 
of the wire to a node where the solder ball should 
be placed. 


1-165. Theoretically, the resonant lengths 1,, Ij, 
1:„ . . . obey the same clamp-free frequency equation 
that is given for the cantilever clamp in paragraph 
1-158. Experiment, however, has demonstrated 
that somewhat longer lengths are required for 
optimum performance. Normally, the free length 
of the wire is made a quarter-wave section, 1,, in 
the frequency range of 20 to 250 kc, and a three- 
quarter-wave section, L, in the range of 250 to 
1000 kc. For. phosphor-bronze wire, the empirical 
formulas for these distances are: 




Figure 1-85. Solder-cone wire mounting of 
face-shear element 


where: d = diameter of wire in inches (usually 
0.0035, 0.005, 0.006.3, or 0.008 in.), 
f = frequency in kc. 

1-166. After soldering to the crystal, the support- 
ing wires are bent to make them serve as springs. 
One, two, or three bends are carefully spaced and 
directed so that the displacement per unit force 
will be the same for all directions. The ends of 
the wires are then soldered without tension to 
metal rods, or "straights,” which in turn are 
welded to eyelets staked in a mica or bakelite base. 
In mounting small crystal plates, the straights are 
little more than short, metal stubs, but larger crys- 
tals are mounted in "cages” having a mica roof as 
well as a mica base. Figure 1-86 shows the cage 
assembly of a 40-kc length-width flexure crystal. 
The cage is formed by two mica plates at each end, 
and four straights. Besides providing for the 
proper mounting of the straights, the mica plates 
also serve as “bumpers” for the crystal. The inner 
and outer plates limit the horizontal and vertical 


WADC TR 56-156 


64 



section I 

Fabrication of Crystal Units 


displacements respectively, thereby protecting the 
unit from wire or crystal damage in the event of 
severe shock or vibration. The spacing between 
bumper and crystal is normally between 25 and 
30 mils. Where the operating frequency is below 
3 kc, the wavelength is usually sufficiently long for 
the entire wire to be cut to a resonant length, so 
that the soldered junction at the straight can serve 
as the nodal terminal. However, the optimum free 
length of wire becomes increasingly critical as the 
frequency is raised. Solder balls are used to estab- 
lish resonance, but at low frequencies small metal 
disks are threaded on the wire to provide greater 
mass while permitting a precise adjustment to 
the correct position; after adjustment, the disks 
are loaded at the back with the correct amount of 
solder. Better control is obtained at the higher 
frequencies without the disk. The solder weights 
range from approximately 80 milligrams for 8-mil 
wire, for large crystals, to 6 milligrams for 3.5-mil 
wire, for small crystals. 

1-167. The principal disadvantages of the solder- 
cone wire support arise from the effects of the 
solder cone upon the electrical characteristics of 
the crystal. To provide a junction of given me- 
chanical strength, a certain quantity of solder is 
required. The solder, however, considerably in- 
creases the effective resistance of the crystal cir- 
cuit as the temperature becomes high; if a high 


crystal Q at high operating temperatures is re- 
quired, the solder cone must be small, and, conse- 
quently, the crystal unit cannot be as rugged 
mechanically as would otherwise be possible. Con- 
versely, if the crystal unit is to withstand severe 
mechanical vibrations and high operating tem- 
peratures, the solder cone must be of maximum 
size, so that the Q and frequency stability will 
necessarily be at a minimum. Furthermore, as the 
volume of solder is increased appreciably, the tem- 
perature-frequency characteristics of the crystal 
may be considerably changed. Normally, the tend- 
ency will be for the zero temperature coefficient 
to shift to a lower temperature ; in extreme cases, 
the zero point may be lost altogether. The tempera- 
ture-frequency effects of hooked wire are generally 
more pronounced than those of straight wire, 
when equal volumes of solder are used. Another 
consideration is the difficulty experienced in mak- 
ing two cones of the same dimensions. 

Headed-Wire Support 

1-168. The headed-wire support (see figure 1-87) 
was developed to obviate the disadvantages of the 
solder cone, while preserving all the advantages 
of the wire type of mounting. The head of the 
wire, which resembles that of a common pin, has 
a diameter of approximately 22 mils for 6-mil 
wire. It is pretinned, and a small globule of solder 
is left at the end for sweating to the crystal ; the 



FACE FLEXUAE 


Figun 1-86. Capo assembly for soldor- 
cone wire mownfinp of low-fraquonty 
fongth’Width flexure crystal * 


volume of solder varies from 1000 to 7000 cubic 
mils, according to the size of the crystal. Phos- 
phor-bronze wire is used, and all other mounting 
details are substantially the same as those for the 
solder-cone type of support. 

1-169. The headed-wire is superior to the solder- 
cone mounting, because it provides a greater and 
more uniformly distributed mechanical support 
with a smaller quantity of solder; in the case of 
low-frequency crystals, the Q is improved by as 
much as twenty-five percent. Furthermore, the 



Figure 1-87. Headed-wire crystal support * 


WADC TR 56-156 


65 



Section I 

Fabrication of Crystal Units 

distance d (figure 1-87) is a constant for all crys- 
tal units of the same design, so that the resonant 
free length of the wire can be predetermined ac- 
curately, thus permitting smaller tolerance in the 
rated characteristics. An additional advantage is 
that the headed wire diminishes the mechanical 
coupling between the vibrating systems repre- 
sented by the crystal and the wires. Standing 
waves are caused, not only by reflections between 
solder ball and crystal, but also to a certain extent 
by reflections from one solder ball, through the 
crystal, to the solder ball on the opposite side. By 
reducing the coupling between crystal and wires, 
the impedance effects due to the interfering 
through-waves are reduced, and a purer frequency 
spectrum is possible. Headed wire may be used to 
replace any other type of low- and medium-fre- 
quency crystal mounting, and a well-designed 
headed-wire crystal unit will generally surpass the 
other types in all-round performance. However, at 
the higher frequencies a clamped air-gap holder 
is still to be preferred for greater activity and 
frequency stability, and at low frequencies, ulti- 
mately the cantilever clamp may prove superior 
for genera] use. 

Edge-Clamped Mounts 

1-170. Two variations of the edge-clamped type of 
mounting are illustrated in figure 1-88. The mount 

EFFECTIVE ELECTRODE 




Figure 7-88. Cdge-clamped systems of mounting. (A) 
Mounting for low-frequeney crystals. (B) Cemented- 
lead mounting for high-frequency crystals 


shown in (A) has been used with low-frequency 
crystals vibrating in extensional or flexure modes ; 
the mount in (B), known as the cemented-lead 
mount, is widely used as an alternate to air-gap 
holders in mounting high frequency, thickness- 
shear elements. Although edge-clamped mounts 
have been successfully used in the production of 
high-activity crystal units for both high- and low- 
frequency applications, this type of mounting 
when used with low-frequency crystals, is probably 
somewhat inferior to well-constructed headed-wire 
or resonant-pin supports. However, a special fea- 
ture of the edge-clamped mounting system is the 
method of dimensioning the electrodes (a method 
also adaptable for use with resonant pins), by 
which optimum performance characteristics can 
be obtained with high-frequency crystals. Plated 
electrodes (or metal foil cemented to the crystal) 
are used, but, as shown in figure 1-88 (B) , the crys- 
tal faces are only partially plated, and the plating 
on opposite faces is extended to opposite edges 
only, so that the effective electrode area is concen- 
trated within a small circular region at the center 
of the crystal. By this means the capacitance is 
kept small, and the principal activity is confined to 
the central region, where the crystal is most likely 
to be of uniform thickness. Both of these factors 
are advantageous in improving the frequency sta- 
bility. Also by reducing the activity in the vicinity 
of the edges, much of the damping due to the im- 
pedances of the supporting structure is obviated. 
Mechanical support and electrical connection is 
supplied by tinned, high-quality spring piano wire, 
which is clamped and cemented to the crystal at 
the edge where electrical contact can be made with 
the lead-outs from the electrodes. The cementing 
is used principally for the purpose of insuring 
good electrical connection, and not for supplying 
mechanical support, which should be provided by 
spring-wire clamps. The base ends of the spring 
wire are coiled around and soldered or welded to 
the base stubs. Although the supporting wires are 
not designed to be resonant elements of the crystal 
unit, they do provide the protection against shock 
and external vibration afforded by the other types 
of wire mounting. As compared with the per- 
formance of fundamental-mode, thickness-shear 
crystals, such as elements A and B, that are 
mounted in corner-clamped air-gap holders, the 
performance of the same elements, when wire- 
mounted, will generally be superior. In addition, 
the wire mounting permits the use of smaller 
crystal holders. The elimination of the air gap re- 
duces the likelihood of arcing, but this does not 
mean that the wire-mounted units can be operated 


WADC TR 56-156 


66 



at higher voltages than the conventional air-gap 
crystal units. This is because the wire-mounted 
crystal is more isolated thermally and tends to be- 
come hotter. The advantages of the cemented-lead 
over the ceramic-button mounting system are less 
pronounced than the advantages over the other 
air-gap systems. For operation at frequencies from 
1 to 10 me, the wire-supported crystal usually has 
the better operating characteristics. As the fre- 
quency increases, however, the metal plating of the 
wire-mounted element becomes an increasingly 
greater factor in damping the oscillations ; and in 
the upper very-high-frequency range, above 100 
me, non-plated crystals that are pressure-mounted 
between ceramic buttons are definitely to be pre- 
ferred. Even in the fundamental frequency range, 
the ceramic-button mounts, which provide the 
better mechanical protection, may be used with 
good effect, and optimum performance character- 
istics for given operating conditions might better 
be achieved by combining the merits of wire- 
mounted edge clamps with those of plated dielec- 
tric buttons. 

HOUSING OF CRYSTAL UNITS 

1-171. The principal function of the housing is to 
provide a hermetically sealed, moisture-resistant 
container. Plastic housings of sandwich, air-gap, 
and clamp-type holders are normally sealed with 
neoprene gaskets. Natural rubber is not recom- 
mended, as the sulphur used in processing the 
rubber will ultimately contaminate other parts of 
the holder. Wire-mounted units are readily adapt- 
able for housing in metal or glass tubes, employing 
standard radio parts; however, small, two-pin 
holders are generally preferred. Before sealing, a 
crystal unit is exposed to high temperature in an 
evacuated oven, in order to drive off adsorbed 
gasses. The sealing itself is usually performed in 
dry air, although certain crystals, particularly the 
flexure-elements, are sealed in vacuum. Optimum 
performance is obtained when a crystal is mounted 
in an evacuated container, since the damping effect 
of the air is eliminated. 

1-172. If metal, rather than glass, housing is em- 
ployed, it is difficult and expensive to seal a Crystal 
unit so perfectly that not even minute leaks will 
develop due to stresses on the pins and the glass- 
sealing of the eyelfts. For this reason most crystal 
units are sealed in dry air, so that if very small 
leaks are present, the crystal characteristics will 
not be appreciably affected for a long period of 
time. Leakage is minimized if the base is rigidly 
protected against deformation, and if the glass 


^ Section I 

Fabrication of Crystal Units 

sealing fills the entire eyelet cavity uniformly. 
However, if a crystal is to be mounted in vacuum, 
a glass housing is to be preferred. 

AGING OF CRYSTAL UNITS 

1-173. “Aging” is a general term applying to any 
cumulative process which contributes to the de- 
terioration of a crystal unit and which results in 
a gradual change in its operating characteristics. 
There are, of course, many interrelated factors in- 
volved in aging — minute leakage through the con- 
tainer, adsorption of moisture, corrosion of the 
electrodes, ionization of the air within the con- 
tainer, wire fatigue, frictional wear, spurious elec- 
trolytic processes, small irreversible alterations in 
the crystal lattice, outgassing of the materials com- 
posing the unit, over-drive, presence of foreign 
matter, various thermal effects, pin strain due to 
socket stresses, and erosion of the surface of the 
crystal. However, if a crystal unit is well designed 
and carefully constructed, the rated operating 
characteristics may well outlast the equipment in 
which the crystal is used. 

1-174. Usually the first effects of aging can be 
traced to changes at the surface of the crystal. 
These changes may be due directly or indirectly 
to almost any combination of the factors mentioned 
in paragraph 1-173, and their occurrence can be 
avoided or greatly diminished only if proper pre- 
cautions and techniques are employed during 
manufacture, and if low driving voltages are em- 
ployed during operation. To produce a crystal unit 
of long life, the final stages of production require 
particular precautions. These concern the finishing 
processes of lapping, etching, cleaning, mounting, 
heat cycling, and protecting against moisture. 

Lopping to Reduce Aging 

1-175. Whether a crystal is being ground with 
abrasives which are cemented or imbedded in a 
grinding disk, or lapped with loose abrasives under 
a lapping disk, the cutting proceeds by virtue of 
the small fractures and chips which result when 
the hard, sharp edges of the abrasive particles are 
rubbed against the surface of the crystal. Com- 
mercial crystals are usually produced by lapping 
with loose abrasives, instead of grinding by 
“grindstones,” except in the initial cutting stages 
and the final edging process, where diamond saws 
are commonly used. Each succeeding lapping stage 
employs a finer grade of abrasive, and must com- 
pletely remove the surface left by the preceding 
stage. The final lapping requires very fine abrasive 
particles, such as 1000- to 1200-grain carborun- 
dum, and should preferably be performed in a 


WADC TR 56-156 


67 



Section I 

Fabrication of Crystal Units 

mixture of abrasive, castile soap, and water. To 
reduce aging, soap and water are preferred as the 
coolant in the finishing stage, rather than kerosene 
or other oils, although kerosene permits a faster 
cutting rate for the same abrasive and lapping 
speed. Apparently, the residue of fractures re- 
maining after a soap-water-abrasive lapping does 
not penetrate as deeply as that remaining after a 
kerosene-abrasive lapping. Regardless of how fine 
the abrasive, small fractures and cracks will be left 
in the surface of the crystal after the final lapping, 
and in time these cracks will spread, absorb mois- 
ture, and ultimately result in a weathering of the 
surface. Additional care must be taken to ensure 



Figure i-89. Minimum change in frequency that AT 
and BT plates must undergo due to etching, if the 
etching is to be sufficient to remove all surface cracks 
and fissions remaining from the final lapping stage. 
Note that, as the crystal becomes thinner, a given 
change in the thickness dimension means a greater 
change in the frequency. The frequency change for a 
BT cut is less than that for an AT cut of the same 
initial frequency, since the larger frequency constant 
of the BT cut permits a thicker plate 


that the crystal is not finished with slight con- 
cavities in the surfaces, or with one end lapped 
down more than the other, making the crystal 
wedge-shaped. Although optimum performance is 
to be obtained with perfectly planar surfaces, 
greater insurance against unwanted non-parallel- 
isms is gained if the lapping is controlled to give 
the plates a symmetrical convex contour of ap- 
proximately 5 microns for lower-frequency crys- 
tals, and approximately lO’/f (cycles) microns for 
crystals above 3 me. 

Etching to Reduce Aging 

1-176. After the final lapping stage, the crystal is 
normally given an etching bath to remove all for- 
eign particles. An eight-minute bath in forty-seven 
percent hydrofluoric acid is sufficient for the aver- 
age crystal, and will permit a firm contact between 
the crystal and its electrode coating. An etching 
time of at least thirty minutes is necessary, how- 
ever, if a minimum aging and a maximum Q, sta- 
bility, and drive level are desired. The longer etch- 
ing period is required to ensure that the deeper 
fissions in the surface caused by the final lapping 
are thoroughly removed. However, the deep etch 
is difficult to control, and particular care must be 
exercised if the desired dimensions of the crystal 
are to be achieved. It is customary to divide the 
deep-etching process into two, or more, steps; 
( 1 ) to etch the crystal to within 1 kc of the desired 
frequency: and (2) in the succeeding steps, to 
bring the crystal within its tolerance limits. Figure 
1-89 indicates the degree of etching required to 
prevent aging in AT and BT cuts. 

CluanlinuM to Ruducu Aging 

1-177. The protection of a crystal from foreign 
matter and moisture is of paramount importance 
if the crystal is to operate with stability and long 
life. Only minute traces of dirt, dust, or finger- 
prints on a crystal will cause the performance to 
be erratic. Cleanliness is necessary throughout the 
final production period, but particular emphasis is 
required during the stages immediately prior to 
sealing. Before and after etching, each crystal 
blank should be scrubbed thoroughly in soap, or 
trisodium phosphate, and water with a soft brush ; 
rinsed in 0.5 percent ammonium hydroxide solu- 
tion, and again washed thoroughly in running 
water; dried in an oven heated to 100 degrees 
centigrade, or in a warm, clean, air stream ; 
washed again in distilled carbon tetrachloride or 
other solvent; rinsed in hot distilled water; and 
finally, carefully dried in an oven. The electrodes 
and holder must be similarly cleaned, and neoprene 


WADC TR 56-156 


68 




tweezers should be used in handling the parts dur- 
ing the hnal assembly. If the crystal is to be metal- 
plated, the complete mounting must be cleaned 
again before sealing. A hot spray of distilled tri- 
chloroethylene for one-half minute is sufficient. The 
plated crystal will normally require a small amount 
of edge-grinding with fine emery paper to bring 
the mounted unit to the proper frequency. This 
step unfortunately weakens the aging resistance of 
the treated surfaces at a stage when further etch- 
ing is no longer feasible for commercial crystals. 
However, before testing and sealing, a retouched 
crystal unit should be thoroughly washed and 
scrubbed, with every precaution taken to ensure 
that no foreign matter remains on the crystal or 
mounting. Where the facilities are available, clean- 
ing can be performed by exciting the bath with 
supersonic acoustic waves, which can clean the 
crystal by shaking all loose fragments off its sur- 
face. In fact, a supersonic bath can be quite as 
effective as an etching bath in reducing aging. 

MouNtiag to Reduce A9in9 

1-178. As a general rule, any deviation in the 
mounting which causes an increase in the fric- 
tional losses will shorten the useful life of a crystal 
unit. Thus, in the nodal types of mounting, small 
deviations from the nodal point in the position at 
which a crystal is held will shorten the life of the 
crystal. Wire-mounted crystals require additional 
precautions during fabrication to avoid local 
changes or stresses at the surface of the crystal. 
Particular care must be taken to avoid electrical 
“twinning," which will occur if the temperature 
is raised above the inversion point, 573°C, and 
then lowered again ; or twinning may be induced at 
a much lower temperature if a sharp temperature 
gradient is present in the crystal. These precau- 
tions are necessary during the baking of the silver 
spots, the division of the electrode coating by elec- 
tric stylus, and the soldering operation. In baking 
the silver spots, the temperature should be kept 
forty to fifty degrees centrigrade below the in- 
version point, and care must be taken to make 
certain that the crystals are heated uniformly. 
Some twinning is inevitable when using an electric 
stylus to divide an electrode coating; however, if 
straight-line division is required, the twinning 
may be avoided by using an abrasive tool or sand 
blasting in place of the stylus. To avoid thermal 
stresses during the soldering operation, a heated 
support should be provided for heating the crystal 
uniformly to a temperature of approximately 
100°C. Twinning, regardless of its cause, primarily 
affects the steady-state electrical characteristics of 


Section I 

Fabrication of Crystal Units 

a crystal element, and only indirectly contributes 
to gradual changes in the performance of the crys- 
tal. At least, no statistical data has been collected 
to show a correlation between twinning and aging; 
nevertheless, a series of small twinned spots at the 
surface is likely to make the area more susceptible 
to erosion. Readjustments of the crystal lattice at 
the twinning boundaries after long periods of elec- 
trical, mechanical, and thermal stresses might be 
expected ; but if these are due to occur, they can 
probably be made to take place by a process of 
artificial aging before the crystal is placed into 
operation. Twinning, however, raises the induc- 
tance and effective resistance of an element, and 
hence, decreases its activity for a given operating 
voltage. Since the ultimate requirement of a higher 
operating voltage can lead to a shortening of the 
life of the crystal unit, an undue amount of twin- 
ning indirectly becomes a factor in the aging. 
Twinning will also raise or lower the frequency, 
according to the particular type of element. If the 
twinning is introduced during the final stages, this 
may require a substantial amount of edge-grinding 
during the final frequency-adjustment stage, and 
more of the etched surface may need to be re- 
moved than otherwise. Thus, although "heat” 
twinning is considered primarily in connection 
with its immediate effect upon the characteristics 
of the crystal, it should also be avoided as an in- 
direct factor in aging. A more direct factor in 
shortening the life of a wire-mounted crystal unit 
is a nonuniformity in the soldered junction, which 
is more likely to occur in a solder-cone than in a 
headed-wire support. When the stresses are un- 
evenly distributed, the soldered junction itself will 
tend to age ; and even if mechanical breakage does 
not occur, the changes in the electrical characteris- 
tics will lead to poor performance and instability. 
Special care must be taken to make certain that 
the silver spots are of uniform density. The con- 
tainers of liquid silver should be agitated for sev- 
eral hours immediately prior to application. Also, 
the critical nature of the soldering operation re- 
quires the aid of a machine and accessories of 
special design. 

Heat Cycling to Reduce Aging 

1-179. A newly mounted crystal will normally ap- 
pear to age more rapidly than one that has been 
in operation for a long period of time. This effect 
is not due to an actual deterioration of the crystal 
unit, but merely to an initial adjustment of the 
crystal, particularly at its surface, to its operating 
environment and changes in temperature. The 
stabilization period can be reduced to one of very 


WADC TR 56-156 


69 


Section I 

Electrical Parameters of Crystal Units 

short duration by subjecting the crystal unit to a surface of the crystal and the corrosion of the 

series of slow heating and cooling cycles varying electrodes. For optimum performance and long 

between 24°C and 116°C. Metal-plated elements life, every precaution must be taken to ensure that 

are frequently heat-cycled during the final fre- the interior of the sealed crystal unit is as free as 

quency-adjustment period, and again after sealing. possible from moisture. Prior to sealing, all com- 

In a series of tests at the Hunt Corporation, it ponents of the crystal unit should be heated in 

was found that negative aging (frequency de- vacuum to drive off absorbed water vapor and 

creases with time) is generally due to insufficient other gases; and if the sealing is performed in 

cleaning of the crystal unit. When this was rem- air, the atmosphere should not have a relative 

edied, it was found that the crystal units would humidity higher than 5 percent, 

then age positively. The cause of the positive 

aging was traced to the outgassing of the metal Low Drive Level to Reduce Aging 

plating of the crystal, and its elimination has been 1-181. As a general rule, the lower the drive level, 

achieved by pre-aging the plated crystal for 3 the longer will be the useful life of a crystal unit, 

minutes in a 300°C oven. After a sufficient period This is true because the cumulative effects of al- 

of artificial aging, a properly fabricated and oper- most all of the previously discussed aging factors 

ated crystal unit will maintain its final tempera- are considerably more pronounced when the crys- 

ture-frequency characteristics indefinitely. tal is operated at high drive levels. Also, the higher 

Low Relative Humidity to Reduce Aging operating voltages greatly increase the tendency 

1-180. A low relative humidity is of paramount toward corona discharge and other ionization 

importance if excessive aging is to be prevented. effects, and the vibrations of greater amplitude are 

Even if a crystal is perfectly mounted and clean, more likely to result in crystal or wire fatigue. To 

an ambient relative humidity higher than 40 per- ensure maximum lifetime, a piezoelectric reson- 

cent will sharply increase the insulation resistance, ator should be driven at the lowest practicable 

and will greatly accelerate the weathering of the level consistent with the circuit requirements. 

ELECTRICAL PARAMETERS OF CRYSTAL UNITS 

EQUIVALENT CIRCUIT Or CRYSTAL UNIT trode parts that extend beyond the quartz. Ch. and 

Ch 2 represent the distributed capacitance of the 
1-182. A crystal unit may be represented by the crystal circuit to the holder H. Ca represents the 

equivalent electrical circuit shown in figure 1-90. capacitance between the electrodes and the crystal 

Ri represents the terminal-to-terminal r-f insula- faces when they are separated by an air gap or 

tion resistance of the crystal unit. Cl, Ll, and Rl other dielectric. If a dielectric exists on both sides 

represent, respectively, the distributed capaci- of the crystal, Ca would equal the total capacitance 

tance, inductance, and resistance of the electrical of the two capacitances in series. Thus, if the air- 

leads and terminals of the mounted crystal. Cl, in gap capacitances on the opposite sides of the crys- 

addition, includes the capacitance across any elec- tal were equal, as would normally be the case, Ca 



WADC TR 56-156 


Figure 1-90, Equivalent circuit of crystal unit 
70 




would be equal to one-half the value of either one. 
C. is the electrostatic capacitance across the quartz 
plate, where the quartz serves as the dielectric. 
The series LCR branches represent the piezoelec- 
tric properties of the crystal as they appear to the 
external circuit when the crystal is undergoing 
mechanical vibrations. For this reason, these 
values are called the “motional-arm” (also, "series- 
arm”) parameters, in contradistinction to the 
parameters such as C« that are not of piezoelectric 
origin. 

1-183. The motional-arm values of L are closely 
associated with the mass of the crystal, those of C 
are closely associated with the elasticity of the 
crystal, and the motional-arm values of R indicate 
the tendency of the crystal to dissipate heat during 
vibration. Each of the motional-arm branches is 
associated with a different mode or harmonic of 
vibration, and the normal frequency of each of the 
modes coincides with the series-resonant frequency 
of the respective LCR branch. It will be assumed 
that the branch indicated by L„ Ci, and R, repre- 
sents the equivalent circuit of the desired mode, 
and that all of the higher subscript branches Lk, 
Ck, Rk> represent unwanted modes. 

1-184. Since a crystal unit is normally intended for 
use only within a very narrow frequency range 
centered at a specified nominal frequency, the 
equivalent circuit may be greatly simplified to that 
shown in figure 1-91. If the crystal is mounted so 
that the electrodes are in direct contact with the 
crystal faces, Ca will not be effective, and the 
values of L, C, and R in figure 1-91 will normally be 
approximately the same as those of L„ Ci, and R, 
in figure 1-90, and C„ will approximately equal 
Chi Chj 

Ce -|- Cl H . For these assumptions to 

Chj -|- Chj 



Section I 

Electrical Parameter* of Crystal Units 

hold, Ri must be much greater than the impedance 
of the crystal when parallel resonance is estab- 
lished between the motional arm and C«. Also, the 
operating frequency must not be so high that the 
reactance of Ll becomes significant ; and the nor- 
mal frequencies of all the unwanted modes must be 
sufficiently removed from the nominal frequency, 
if each of the unwanted branches is to present a 
high impedance at the desired operating frequency. 

SIMPLIFIED EQUIVALENT CIRCUIT OF 
AIR-GAP CRYSTAL UNIT 

1-186. C, is normally much greater than the dis- 
tributed capacitance across the leads, so an air- 
gap or dielectric-sandwich type of crystal unit may 
be represented by the equivalent circuit shown in 
figure 1-92. This circuit, in turn, may be reduced 
to the equivalent circuit of figure 1-91 by assigning 
the following values to L, C, R, and C„ : 

C = Ca^C, 

(Ca + Ce) (Cl Ce Ca) 

CACe 

“ Ca -f Ce 

THE EFFECT OF R-F LEAKAGE RESISTANCE 

1-186. The principal effect of Ri, the terminal-to- 
terminal r-f leakage resistance shunting the crys- 



Flgur* t-91. SImpfiRed »tfulvalant elreuH of 
crystal unit 


Figura 1-97. Slmplifiad equivalent circuit of air-gap 
crystal unit 


WADC TR 56-156 


71 


section I 

Electrical Parameter* of Cryatol Units 

tal, is to reduce the effective Q of the crystal unit. 
For all practical purposes this effect is negligible 
when the crystal is being operated at, or very 
near, the resonant frequency of the series arm. 
Under these conditions the electrical impedance of 
the crystal is so small by comparison that Ri can 
be ignored. On the other hand, as the frequency 
rises above the resonant point, the impedance in- 
creases sharply, and the greater the impedance be- 
comes, the greater is the effect of a given Ri. 
Insofar as the equivalent circuit of figure 1-91 is 
concerned, the effect will be to increase the value 
of R. The extent of this increase will depend upon 
how large the effective reactance of the crystal be- 
comes, relative to Ri. For the sake of simplification, 
most of the discussion given later concerning the 
equivalent circuit assumes that the increase in R 
due to Ri is negligible, or at least, is constant, re- 
gardless of the frequency, an assumption that can 
produce reasonably accurate results in the case of 
well-fabricated crystal units. The leakage resist- 
ance of military crystal units has a specified mini- 
mum d-c value of 500 megohms. As long as this 
minimum d-c value is maintained, R, at low fre- 
quencies will be comparable to this value. However, 
if an accumulation of moisture, dirt, or the like 
seriously reduces the d-c insulation resistance be- 
low the allowed minimum, the off-resonance char- 
acteristics will undergo a noticeable change. For 
instance, low-frequency filter crystals may have 
impedances at antiresonance in the neighborhood 
of 50 to 100 megohms. If Ri decreases below 500 
megohms, the equivalent R of the motional arm 
will increase markedly. In the case of high-fre- 
quency crystal units, the effective dielectric losses 
may become relatively large, particularly when 
plastic holders are used, so that, at off-resonant 
frequencies, R, can become a significant parameter 
of the over-all effective resistance. However, for 
high-frequency crystal units employing modern 
methods of mounting and construction, R, can gen- 
erally be ignored. In the very-high-frequency 
range, crystal units are almost always operated at 
series resonance, so that, even if R, were on the 
order of 100,000 ohms, as might easily be the case, 
the effect would still be relatively minor. However, 
where the shunt resistance cannot be ignored, a 
more concrete analysis of its effect is to let R,, in 
figure 1-90, represent only the d-c leakage resist- 
ance, and to account for the r-f dielectric losses by 
inserting other equivalent resistances in series 
with the various shunt capacitances. In the sim- 
plified equivalent circuit in figure 1-91, the d-c 
leakage resistance could still be ignored, but non- 
negligible r-f shunt losses could be interpreted as 

WAOC TR 56-156 


being due to a single resistance in series with the 
shunt capacitance, Co- The Q of the equivalent 
shunt arm will effectively equal the Q of the crystal 
unit when the unit is operated at frequencies well 
removed from resonance. The crystal units that 
are mounted in metal or glass holders of the type 
described in Section II, and are recommended for 
use in equipments of new design, can be expected 
to have shunt-arm Q’s greater than 1000 at all 
frequencies within their specified range. This as- 
surance, however, cannot be given for the crystal 
units mounted in plastic holders, particularly the 
old-style phenolic holder, or for those employing 
all-metal sandwich or air-gap electrodes. However, 
the lower Q’s of the older types of crystal holders 
are not entirely due to greater dielectric losses and 
larger values of shunt capacitance. An equally im- 
portant factor is the effective inductance of the 
circuit effectively in series with the shunt capaci- 
tances. For example, a corner-clamped air-gap 
mounting, such as that provided in a DC-31 crystal 
unit, has an effective shunt-arm Q of approxi- 
mately 80 or 180 at 30 me, depending upon whether 
the clamping pressure is applied by a coiled or a 
flat spring, respectively. Apparently, the reactance 
and resistance of a coil spring can be quite detri- 
mental to the quality of a crystal holder at very 
high frequencies, since it can cause not only an 
effective increase in the shunt capacitance, but 
also an increase in the effective dielectric losses. 
These losses would become prohibitive if the in- 
ductance of the spring and its stray capacitance 
should approach the properties of a series-resonant 
arm shunting the crystal. However, except in such 
abnormal cases, and in cases where the insulation 
is weakened by extremes in humidity and tempera- 
ture, the shunt resistance will have a negligible 
effect upon the performance of a crystal circuit. 

EFFECT OF DISTRIBUTED INDUCTANCE 

1-187. The effective self-inductance of the crystal 
leads, Lu is normally not sufficient to seriously 
affect the crystal parameters, except in the case 
of very high operating frequencies where it is 
necessary to operate the crystal at series reson- 
ance. At resonance, the reactance of the crystal 
unit will be zero, so that the crystal, in combination 
with its shunt capacitance, must have a net equiva- 
lent series Xc equal in magnitude to the Xll of the 
distributed inductance. This means that the reson- 
ant frequency will be slightly lower than would be 
the case if there were no distributed inductance. 
The net effect on the equivalent circuit of figure 
1-91 is that the LC product is increased very 
slightly (lower resonant frequency), and that C, 


72 



is increased to a greater extent. If the distributed 
inductance is completely negligible, the resonant 
frequency of the crystal will be slightly higher 
than the normal resonant frequency of the series 
arm, because of the reactive component of current 
through Co. However, the distributed Xl of the 
lower-frequency crystal units may be sufficient to 
approximately cancel the reactance due to the true 
Co. Under these conditions, the resonant frequency 
of the crystal unit as a whole would coincide with 
the natural vibration frequency of the crystal — an 
ideal operating state. In the case of the higher- 
frequency crystal units, the distributed inductive 
reactance may be sufficient to lower the frequency 
below the natural resonance point by several cycles. 
If the crystal unit were being operated at series 
resonance in a capacitance-bridge circuit, for ex- 
ample, such an effect would lead to frequency 
jumps with slight changes in the tuning adjust- 
ments. Under such conditions it would be desirable 
to add a capacitance in series with the crystal, 
with a reactance just sufficient to cancel the un- 
wanted Xll- The distributed inductance, Ll, of the 
lower-frequency crystals, and of practically any 
crystal unit which is to be operated above series 
resonance, has only a minor effect. The maximum 
effect will always be at very high frequencies near 
series resonance. In analyzing the behavior of a 
crystal unit where the distributed Xll cannot be 
neglected, the simplest approach is to consider Xi, 
as a separate fixed reactance in series with the 
crystal unit. From this point of view, as long as 
Xll is very small, as compared with Xo, it can be 
seen that Li, will not seriously affect the rate at 
which the net crystal reactance will change with 
frequency, and, therefore, will not influence the 
stabilizing effect of the crystal on the frequency. 
Crystal oscillators can operate successfully up to 
frequencies as high as 200 me. However, crystal- 
control of the frequency can be stable only when 
the impedance at series resonance is much smaller 
than the reactance of the effective shunt capaci- 
tance C„. The larger the value of Xu., the smaller 
this ratio will be. Thus, the higher the frequency, 
the greater the importance of keeping the crystal 
leads as short as possible, not only to reduce Ll, 
but also to reduce the distributed capacitance and 
the r-f resistance of the wires. The small coaxial- 
electrode type of mounting, such as the HC-IO/U, 
is the most satisfactory for achieving a minimum 
effective Co, and hence, a maximum frequency 
stability in the very-high-frequency range. It 
should be remembered, however, that since the dis- 
tributed Xll will increase with the frequency, the 
effective Co will also increase with the frequency. 


S*cKon I 

Electrical Parameters of Crystal Units 

SO that a measurement of C, at a frequency far 
lower than that of resonance will not alone give a 
reliable indication of the effective parameter near 
the operating frequency. 

EFFECT OF DISTRIBUTED CAPACITANCE 

1-188. The effect of the distributed capacitance 
on the parameters of the simplified equivalent cir- 
cuit is merely to increase the value of Co. However, 
it should be noted that the amount of the increase 
will depend somewhat upon how the crystal unit is 
connected in the external circuit. For example, 
assume that the holder and terminal 1 in figure 
1-90 are grounded. Chi, which would otherwise be 
in series with Chs, is now effectively short-cir- 
cuited, so that the total shunt capacitance C. is 
increased. If Chi were assumed to be equal to Chi, 
the amount of the increase due to grounding ter- 
minal 1 and the holder would equal Chi/2. On the 
other hand, grounding the holder can result in an 
effective decrease in C„. Assume, for instance, that 
a crystal unit is connected in a circuit equivalent 
to that shown in figure 1-93. With the metal holder 
ungrounded. Cm and Chi are effectively connected 
in series, so that, if Ch. = Chi, the total capaci- 
tance of the series combination is Ciii/2. If switch 
S is closed, thereby grounding the holder, the effec- 
tive total Co becomes larger or smaller, depending 
upon the point of view of the observer. Since Chi 
is no longer in series with Chi, but, instead, is 
shunted across the entire circuit, whereas Cm is 
shunted across the load Z, the total capacitance 
facing the generator is increased (assuming that 
Z is the reactance of a capacitor). When S is 
closed, the current through M, increases; how- 
ever, the current through Mj decreases. An ob- 
server at Ml would say that grounding the holder 
increased Co, whereas an observer at M, would 
say that Co has decreased. At frequencies well re- 
moved from the nearest resonant frequency of 
the motional arms, the branch impedances are so 



Flgura 1-93. Cirevit diagram Indicating the effect that 
grounding a motal holder may have on shunt 
capacitance 


WADC TR 56-156 


73 





Section I 

Electrical Parameters of Crystal Units 

high that the crystal unit behaves essentially as a 
capacitor of value C,. It can be seen that if a meas- 
urement were being made of the change in C„ due 
to the grounding of the holder, it would be impor- 
tant to know exactly how the measurements were 
made. For example, low-frequency crystal units 
mounted in the HC-13/U have been reported as 
having 0.8 ix/if less shunt capacitance, and medium- 
frequency crystal units mounted in the HC-6/U 
holder have been described as having 0.5 jn/if less 
shunt capacitance with the holder grounded. How- 
ever, it should be noted that crystal units so speci- 
fied are intended primarily for use in circuits 
where the crystal operates in a series-resonant 
rather than a parallel-resonant circuit. Even so, 
the grounded holder alters the entire circuit, not 
simply Co. Thus, in the circuit of figure 1-93, sup- 
pose that it is necessary for the current through Z 
to be in phase with the generator voltage. If the 
circuit is properly adjusted with an ungrounded 
holder, grounding the holder will detune the cir- 
cuit by effectively decreasing Co, on the one hand, 
and on the other, by shunting the load with Ch 2 . 
The over-all effect cannot be predicted simply by 
specifying an effective change in Co, since the end 
result will depend upon the impedance character- 
istics of the entire circuit. If the frequency of a 
crystal oscillator is being measured by beating its 
output with the output of a frequency standard, 
it is common practice to touch the crystal holder 
with the hand in order to determine whether the 
crystal unit which is touched is operating at a fre- 
quency above or below that of the standard oscil- 
lator. The oscillator frequency will be higher or 
lower than that of the standard according to 
whether the hand capacitance causes the beat fre- 
quency to fall or to rise, respectively, provided 
that the effective C„ is actually increased by the 
touch of the hand, as is invariably assumed. Before 
this assumption is made with complete assurance, 
however, the response of the circuit to a grounded 
holder should be known. 

EFFECT OF DISTRIBUTED RESISTANCE 

1-189. Rl is assumed to include only the ohmic re- 
sistance of the electrical leads and the reflected 
resistance due to eddy currents in the holder and 
ground connections. At normal frequencies Rl is 
quite small, as compared with Ri; even the small- 
sized supporting wires of wire-mounted crystals 
have r-f resistances that are measurable in tenths 
of an ohm. As in the case of the other distributed 
parameters, the effect of Rl upon the equivalent 
circuit of figure 1-91 becomes more pronounced at 
the higher frequencies. To a first approximation 


R is simply R, -|- Rl. However, at frequencies 
above 10 me, the r-f resistance of the leads in- 
creases directly as the square root of the fre- 
quency, so that, in the v-h-f range Rl may be 
greater than one ohm. Rl will also increase some- 
what if the holder is grounded, as the increased 
eddy-current losses in the shielding will be re- 
flected as additional resistance losses in the crystal 
circuit. At normal frequencies, however, the effect 
of Rl is of minor importance; and even at fre- 
quencies above 100 me, its consideration is sec- 
ondary to the effects of the distributed capacitance. 

RULE-OF-THUMB EQUATIONS FOR 
ESTIMATING PARAMETERS 

1-190. The crystal parameters for a given fre- 
quency vary rather widely from one crystal unit 
to the next. Even crystal units of similar dimen- 
sions and fabrication made by the same manufac- 
turer may show significant differences between 
corresponding parameters. These differences arise 
from the sensitivity of the quartz plate to slight 
changes in its angle of cut, surface state, elec- 
trode area, soldered connections, and the like. The 
parameter with the greatest percentage variation 
is R, and it is not uncommon for the larger values 
of R to be from 300 to 900 percent greater than 
the minimum values. The most predictable param- 
eter is C„ since it is primarily a linear function 
of the electrode area, the thickness of the quartz 
dielectric, and the dielectric constant, all of which 
are reasonably constant for a given fabrication 
technique, although variations may be expected in 
crystal units of the same nominal frequency and 
type of mounting, when made by different manu- 
facturers. For the same manufacturer, nominal 
frequency, and type of crystal unit, however, C, 
rarely varies by more than ±6% of its nominal 
value. With a reasonably constant as a starting 
point, approximate values for the major param- 
eters L, C, R, and C„ may be predicted for the 
principal types of crystal elements and holders. 
First, Ce is computed from the known values of 
plate area, dielectric thickness, and dielectric con- 
stant. Next, C can be found, since it is theoretically 
equal to C,. times a constant of proportionality. L 
can next be computed, since the LC product must 
conform to the nominal frequency. Next, an ap- 
proximate range of the values of R may be esti- 
mated from the empirical values of the crystal 
quality factor, Q. Since Q is the ratio Xl/R (or 
— Xc/R), R is thus equal to Xl/Q. Finally, C„ can 
be estimated by simply adding to C* the approxi- 
mate total distributed capacitance common to the 
particular type of holder and mounting. 


WADC TR 56-156 


74 



Estimating C„ Static Capacitance of Crystal 

1-191. Althougrh the dielectric constant of quartz 
varies somewhat according to the angle of cut, 
the following formula will be approximately cor- 
rect for plated electrodes : 

C. = 0.402 A/t ixfit 1—191 (1) 

where A is the effective electrode area in square 
centimeters, and t is the thickness in centimeters. 
1-192. In the case of partially plated A elements, 
where t is a function of the nominal frequency and 
the harmonic, equation 1-191(1) may be expressed 
as: 


Ce = 2.42 Af/n 


1—192 (1) 


where f is the nominal frequency in mc/sec, and 
n, an odd integer, is the harmonic of the thickness- 
shear vibration. Although the quartz plates range 
from 1 to more than 2 sq cm in plate area, the 
electrode area normally covers only a fraction of 
the total quartz surface. The RTMA Standards 
Committee on Quartz Crystals has recommended 
the following approximate electrode areas for the 
fundamental frequencies of this type of crystal 
unit. 


Frequency in mc/sec 
(n = l) 

1 — 2 
2 — 5 
5 — 9 
9 — 15 
15 — 20 


Electrode Area ±10% 
(sq cm) 

0,504 

0.385 

0.283 

0.159 

0.126 


For the overtone modes, where n is greater than 
1, the electrode area will be the same as that of 
the fundamental mode of frequency equal to f/n. 
The harmonics for various ranges of f are as 
follows : 

f = 10 — 45 me ; n = 3 
f = 45 — 75 me ; n = 5 
f = 75 — 105 me ; n = 7 

1-193. In the case of crystals vibrating in a face- 
shear mode, it is the electrode area A that is a 
function of the frequency. For fully plated C ele- 
ments, equation 1-191(1) may be expressed as: 


Ce = 0.038/tf* pMf 

where t has an average value of 0.05 cm, and f 
(mc/sec) lies between 0.3 and 1 mc/sec. 

1-194. For fully plated D elements, equation 
1 — 191(1) may be expressed as: 


C, = 0.0172/tf^ Mpf 


SecKon I 

Electrical Parameters of Crystal Units 



0 20 40 60 SO ’ 100 

f (kc/sec ) 


Figure 1-94. C, versus frequency for typical wire- 
mounted N elements 

where t has an average value of 0.05 cm, and f 
(mc/sec) lies between 0.2 and 0.5 mc/sec. 

1-195. For a typical wire-mounted J element, 
equation 1 — 191 ( 1 ) may be expressed as : 

Ce = k/f ppf 

where; k =: 38 for f = 1.2 to 2.5 kc/sec 

= 45 = 2.5 to 4.0 kc/sec 

= 58 = 4.0 to 6.6 kc/sec 

= 77 = 6.6 to 10.0 kc/sec 

Note that f in this case is to be expressed in kc/sec. 
1-196. Typical values of C.. for an N element are 
shown in figure 1-94. 

Estimating C, Eguivalent Motionai-Arm 
Capacitance 

1-197. After C, is known, an approximate value 
for C at the fundamental frequency can be readily 
obtained from the following equation : 


C = C„/re 


1—197 (1) 


A ELEMENTS, l/t> 5 2 50 
B ELEMENTS.l/t >5 650 
C ELEMENTS.w/* = I 360 


G ELEMENTS, o/C^.BS 35 0 
J ELEMENTS,! /I <06 2 00 
M ELEMENTS, «/e= .4 190 



Figure 1-95. Approximate values of the ratio of 


capacitances, 


r^ = , for various plated crystal 

W 

elements 


WADC TR 56-156 


75 



Section I 

Electrical Parameters of Crystal Units 

where r, is simply the ratio of the electrostatic 
capacitance C, to the motional capacitance C, with 
C, and C expressed in the same units. The values 
of r^ for the more important elements are given 
:n figure 1-95. For the odd harmonics (n) of the 
thickness-shear modes: 

C = Ce/ren' 1—197 (2) 

Estimatin9 L, Equivalent Motional-Arm 
Inductance 

1-198. Since Xl is equal to Xf at the series-reso- 
nant frequency of the motional arm, L is found 
quite simply, once f and c are known. Thus: 

L = 2^2 „ henries 1 — 198 (1) 

4^ f C/ 

Remember, however, that f is expressed in cycles/ 
sec, and C in farads. 

Estimating R, Equivalent Met!onal-Arm 
Resistance 

1-199. A theoretical equation for R would not be 
practical, since this parameter is much too sensi- 
tive to slight variations during the fabrication 
process and to changes in the crystal drive. An 
approximate estimate is gained from observations 
of the value of Q for the various frequency ranges. 
Thus: 


R = ohms 1—199 (1) 

Q 

where f is in cycles/sec, and L is in henries. The 
values of Q will range from 10,000 to 200,000, 
and in exceptional cases will have much higher 
values. Generally, the higher Q’s are to be found 
at the higher frequencies. For face-shear elements, 
the average Q is approximately 30,000, with most 
values falling between 10,000 and 40,000. Thick- 
ness-shear elements will have average Q’s of ap- 
proximately 75,000, and most of the values will 
lie between 35,000 and 100,000. 

1-200. The Q is not a dependable parameter, and 
will vary from frequency to frequency, and from 
manufacturer to manufacturer, for the same type 
of crystal unit. For example, when' expressed as 

Q = — Xo/R = ^ it can be seen that Q is 

inversely proportional to C, and thus might be 
considerably increased by simply reducing the 
area of the electrodes. On the other hand, the re- 
sistance, R, is at least limited in practice by mili- 
tary specifications. For this reason, the typical 


9000 
8000 
7000 
6000 
5000 
4000 
3000 
2000 
1000 
900 
800 
700 
600 
500 
400 
300 
200 
lOO 
90 
80 
70 
60 
50 
40 
30 
20 
I 0 

c 





















































































































































7 
















- ( 


)K 



















- 
















































































“1 




























































































h 













V, 





L_ 



















3 























n 























n 
























\ 













































1 


























■s 












r 








(n 


r:] 

'7 




!ns 

5) Kc 

»f 

XIO- 



L_ 








-n 

















± 








inrr: 

IT 


— 


— 

— 


Figure 7-96. Typical curves of the serres-arm resist- 
ance of plated crystals versus frequency. Actual 
series-arm resistances will vary between R/3 and 3R, 
where R is the value shown, except when R is less 
than 10 ohms, in which case the minimum resistance 
will be approximately one-half the value indicated. 
Values indicated are average for fundamental modes 
and approximately 'At the average for overtone modes 

values of R versus f, shown in figrure 1-96, are 
more likely to be found in randomly selected crys- 
tal units than is a given value of Q. The values of 
R indicated in figure 1-96 are merely typical, how- 
ever, and a small percentage of actual Military 
Standard crystal units will have series-arm resist- 
ances as small as one-third, or as large as three 
times the amounts showh. 

Estimofing Co* Totoi Static Shunt Copoeitonco 

1-201. The equation for Co is 

Co = Co + Cd 

where Cd is the total distributed capacitance of 
the crystal leads and terminals. Approximate 
values of Cd for plated crystals in ungrounded 
holders are given below : 


Crystal Holder 

Cdiiifif) 

HC-6/U 

0.7 

HC-IO/U 

0.3 

HC-13/U 

1.0 

HC-15/U 

1.5 


WADC TR 56-156 


76 



IMPEDANCE CHARACTERISTICS 
VERSUS FRE9UENCY 

1-202. The superiority of the quartz crystal as a 
frequency stabilizer lies in the fact that a small 
change in the frequency will cause a much larger 
change in the impedance of the equivalent circuit 
than can be obtained with conventional inductor- 
capacitor networks. Where an ordinary r-f tank 
coil would have an inductance measured in micro- 
henries, and an effective Q of 10 to 250, the equiva- 
lent circuit in figure 1-91 will have an inductance 
measured in henries and a Q of 10,000 to 250,000 
or more. C, of course, is extremely small, since its 
reactance must equal Xi, at resonance, and is com- 
monly expressed in milIi-/*/*f (thousandths of a 
micromicrofarad). R is expressed in ohms, and 
although at low frequencies it may have values 
higher than 3000 ohms, depending upon the par- 
ticular crystal element and method of mounting, 
the more common values lie between 10 and 100 
ohms. Co normally lies between 3.5 and 14 nnf, 
although much larger values are encountered 
where electrodes of large surface area are em- 
ployed. Among the smaller holders, such as types 
HC-6/U and HC-IO/U, values of 5 to 6 /i/xt are 
quite common. 

1-203. Since Xi, = 2rrfL 


and Xc = 


-1 

2irfC 


then, the rates at which Xl and Xo change with 
frequency will be, respectively : 


Section I 

Electrical Parameters of Crystal Units 


or 


or 


2Tf.L = 


1 

2Tf.C 


2irL 


1 

2s-f,*C 


However, note that this last equation not only 
implies that the two reactances have equal magni- 
tudes at the series-resonant frequency, f„ but also, 
that f, is the one frequency at which both react- 
ances will change with frequency at the same rate. 
Therefore, for small changes in frequency near 
series resonance: 


AXt, = AXc 

And since the total change in the reactance of the 
series arm is 

AX, = AX|, "1- AXc 

then 


AXg = 2 AXi. = 47rLAf 

If f. is taken as the reference frequency, so that 
af = f — f., then, since X, = 0 at resonance, the 
total reactance of the series arm, X„ will be equal 
to aX,. That is ; 

X. = 4TLAf 1—203 (1) 

Thus, for all frequencies near f„ the equivalent cir- 
cuit of a crystal unit may be represented as shown 
in figure 1-97, where Xc„ and R may be assumed 


dXc ^ 1 

df 2xf"C 

Note that both of these derivatives indicate a posi- 
tive change in reactance with an increase in fre- 
quency. However, it should be remembered that 
Xc is negative, so that a positive change in Xo 
means that its magnitude becomes smaller as the 
frequency increases. On the other hand, the re- 
actance of the inductance increases by an amount 
2irL for each additional cycle per second. At the 
series-resonant frequency of the series arm, the 
total reactance 


Xv+ Xc = 0 



’‘‘=o'"2,fsCo 

Xs * 4 v L Af 


Figure I -97. Impedance diagram of equivalani circuit 
of crystal unit 


77 


WAOC TR 56-156 




section I 

Electrical Parameters of Crystal Units 



x« 


R. 


Xco[R* + X| + XeoX,] 

3 m ' 

R' + [x,^):co]* I , LA» 
Xco 

RXio ^ R 

R' + [x, + Xco]* 


(3) 


= V^R* + Xj 


O 


@ Al = f-f. 


Figure 1-98. Cguiva/ent circuit of crystal unit when 
represented as an effective reactance in series with an 
effective resistance. The ganged arrows indicate that 
Xp and Rp will vary together with changes in the 
frequency, as indicated by the approximate formulas 
given as functions of Af. (Xco is negative.) 

to be constant, but with X, a variable that changes 
linearly with Af, and has the same sign as Af. 
1-204. The series-parallel circuit of figure 1-97 
may be reduced to an equivalent circuit of X, and 
Re in series, as shown in figure 1-98. It should be 
remembered, however, that X(,« is negative, where- 
as X. is either negative or positive, according to 
the sign of Af. The values of Re and X„ expressed 
as functions of Af, are not exact, but are close 
approximations, well within the accuracy of the 
normal test procedure, except when the numera- 
tors reduce to zero. Note, however, that with 
Af =r 0, the approximate expressions equate X, 
to 0, and Rp to R. This is equivalent to assuming 


that Xco is infinite by comparison with R, so that 
at series resonance of the motional arm the crys- 
tal unit as a whole behaves as a pure resistance 
equal to R. Although this is a close approximation, 
it is not exact. For X, actually to be zero, the 
term (R* -f- X,* -)- Xco X,) must be zero. There 
are two frequencies at which this will occur. One 
is called the resonant frequency of the crystal unit, 
fr, and the other is called the parallel-resonant, or 
antiresonant frequency, f,. 

RESONANT FREQUENCY OF CRYSTAL UNIT 

1-205. First, it should be remembered that f„ the 
resonant frequency of the crystal unit, is almost, 
but not exactly, identical with f„ the series-reso- 
nant frequency of the motional arm. If there were 
no shunt capacitance, Co, then f, would indeed be 
the same as f, ; but, as it is, C„ introduces a reactive 
component to the current which must be cancelled 
by a reactive component of opposite phase through 
the motional arm, if the crystal unit is to appear as 
a pure resistance. These conditions are illustrated 
(not to scale) in the vector diagram of the currents 
through the two arms of the crystal unit, shown 
in figure 1-99. ‘Note that the frequency at which 
the crystal unit has the lowest impedance (maxi- 
mum current) is f.. Since X, = 0 at this frequency, 
the equivalent circuit of figure 1-98, according to 

* This sentence applies only to the relative impedances 
suggested by the current vectors in figure 1-99. It can be 
shown that the true minimum impedance of the crystal unit 
occurs at a frequency, f,„, that is as far below f, as f, is 
above f,. 


RE4CT(VEI total CURRENT AT SERIES 

+ RESONANCE (fr) 

I( = total CURRENT AT SERIES 

RESONANCE OF MOTIONAL ARM (f, ) 

= Uo T I»s 

lcg= CURRENT THROUGH Co" CONSTANT 

1., = CURRENT THROUGH MOTIONAL ARM AT f, 

1., 'CURRENT THROUGH MOTIONAL ARM AT (, 
Iir«=-*Co ° reactive COMPONENT OF 

* resistive COMPONENT OF I,, 


I 


sri 



RESISTIVE 


I 


Figure 1-99. Fhasor representation (greatly exaggerated) of current 
through arms of crystal unit at the series-resonant frequencies, f, and f,. 
Distributed inductance of the crystal leads is assumed to be negligible 


WADC TR 56-156 


78 



equations 1 and 2, becomes 


and 



R. = r( 


R^ 


R" + Xc. 


Xr 


R"-|- Xc 


0 

) 


Except at the very high frequencies, Xc„ is much 
larger than R, so that R, «= R, and X^ is so small 
that it may well be more than annulled by the dis- 
tributed inductance of the external wiring. Even 
at frequencies in the neighborhood of 100 me, Xq, 
will have a magnitude in the vicinity of 400 ohms, 
or approximately 10 times or more than that of R, 
so that Re will equal R within ±1 percent. The true 
frequency at which a “series-resonant” crystal 
circuit is intended to operate, however, is f,, where 
all the reactive components of crystal current 
cancel. Actually, the term “series-resonance” is 
somewhat misleading, for the conditions of crystal 
resonance are those of a parallel, and not a series 
circuit. It should be understood that when we 
speak of series-mode circuits and oscillators, the 
operating frequency is generally assumed to be f,. 
1-206. By equation 1, figure 1-98, in order for X, 
to be zero, the frequency must be such that ; 

R* = - (X.^ -I- Xc„ XJ 

Since X(;„ is negative, this equality can only exist 
when X, is positive, i.e., X, is inductive, and f > f.. 
At frequencies very close to f., Xc„ > > X„ so that 
X,’ may be considered negligible. Thus, f^ will be 
the frequency at which 


R* = - Xc„X. = - 4 t LXcoAfr 


where Af, = f, — f. 
Since X^o is negative, 

Af = - R' 

' 4tLXc„ 


1—206 (1) 


Section I 

Electrical Parameters of Crystal Units 

According to figure 1-95, r, = 250. Thus, by 
equation 1 — 197 (1): 

By equation 1 — 198 (1); 

. ^ 10 '^ 

4 X 3.14" X 10’" X 1.54 

= 1.65 X 10 " henries 

According to paragraph 1-201: 

Co = 3.85 -1- 0.7 = 4.55 ixfil 

So that 

X = - 1 = - 10'" 

27rf, Co 6.28 X 10" X 4.55 

= - 3.5 X 10* n 

From figure 1-96, a typical value of R is found to 
be 8 n. On substitution of the foregoing values of 
R, L, and Xro in equation 1 — 206(1) , we find that: 


4 X 3.14 X 1.65 X 3.5 
= 0.088 cycle/sec. 

With such an extremely small difference between 
the two resonant frequencies of the crystal unit 
(less than 1 part of 10*), for all practical purposes 
it can be assumed that f, = f,. Indeed, it would be 
academic to seek to distinguish between them. Re- 
member, however, that the discussion has only 
concerned the equivalent circuit, in which the 
effects of the distributed inductance have been 
assumed to be reflected in a lower series-arm fre- 
quency,- and a larger C*. If the parameters in the 
example above are assumed to be the “true” 
values, so that the inductance of the leads must 
be represented separately, then a slightly more 
realistic interpretation will be possible. Assume, 
for instance, the Li, = ,10-' henries. Then 


1-207. As a concrete example, assume that a par- 
tially plated A element, mounted in an HC-6/U 
holder according to RTMA recommendations, op- 
erates at resonance in its fundamental mode at a 
nominal frequency of 10 me. Approximately, what 
value of Af, could be expected ? Referring to para- 
graph 1-192, we find that A = 0.159 sq cm. On 
substitution in equation 1 — 192(1) ; 

C, = 2.42 X 0.159 X 10 = 3.85 


X^L = 27rL,f, = 0.628 

In order for Xe to cancel this reactance, then, by 
equation 1 in figure 1 — 98: 


- 4jrLAf, = 0.628 


or 


Af, = 


- 0.628 

4ir X 1.65 X 10 " 


= 3 cycles/sec 


WAOC TR 56-156 


79 



Saction I 

Elactrieal Parameters of Crystal Units 

The inductance of the external connections could 
easily increase this value of Af, ten-fold, so that 
for optimum frequency stability, an external series 
capacitance would be necessary. It is important to 
note the negligible effect that a small change in 
R or C« will have on the frequency of a crystal 
unit at series resonance. In equation 1 — 206(1), as 
applied to the 10-mc crystal unit, even if R should 
triple in value, the frequency would not change 
by more than 1 part in 10'. Although the power 
transferred through the crystal would be dimin- 
ished, as would the Q, and hence, the effectiveness 
of the crystal as a frequency stabilizer, a reason- 
able increase in R will not, in itself, cause the fre- 
quency of a series-resonant crystal oscillator to 
drift. 


ANTIRESONANT FRE9UENCY OF 
CRYSTAL UNIT 

1-208. Returning again to equation 1 of figure 1-98, 
it can be seen that the term (R’ -j- X,* Xc® X,) 
can also be zero at some higher frequency than fr, 
namely, when X, = Xo, (R’ being negligible) . This 
would represent the high-impedance, parallel-reso- 
nant state of the equivalent circuit in figure 1-97. 
Letting af, = f, — f, then, at f„ the antiresonant 
frequency 


so that 


X„ = 4t LAf. = I Xco I 


IXcol 

4ir 


1—208 (1) 


On substitution of the typical values of Xco and L 
that were found for the 10-mc crystal unit: 


Af. 


3.5 X 10^ 

4 X 3.14 X 1.65 X 10-* 


16.9 kc/sec 


For a 10-mc crystal, this value of f, represents a 
0.169 percent frequency range in which the crystal 
may be used as a frequency-control device. At all 
frequencies within its range, except at f, and f., 
the unit will appear to the external circuit as an 
inductive reactance, X„ in series with a resistance, 
R,. There is a very simple relation between the 

Af 

fractional frequency range, -j*-, and the ratio of 
(2 

the capacitances, r = ^, that can be derived from 
equation 1. Thus: 

"" 4,rL2irfCo 


so 


Af./f 


1 

2 (/ LCo 


where to = 2 t f. Now, 

01 ^ = 1/LC 


so, on substitution: 

Af./f = C/2Co = 1/2 r 


1—208 (2) 


In the case of plated crystals, r is usually some- 
what less than that predicted by theory. Where 
it should be slightly greater than the values of 
r, in figure 1-95, since C« > C., it is usually some- 
what less. However, as a practical rule-of-thumb, 
it can be assumed that r = r., but only in those 
cases where C. <=» C,. The ratio of capacitances, r, 
is quite an important parameter of the crystal unit 
in its own right, not only as an indication of the 
maximum percentage width of the frequency band 
in which a particular crystal element can operate, 
but, as will be discussed later, as a measure of the 
electromechanical coupling, and also, because of 
its relation to the frequency stability. 


IMPEDANCE CURVES OF CRYSTAL UNIT 

1-209. Figure 1-100 shows the typical characteris- 
tics of the equivalent impedance circuit of figure 
1-98, but with the frequency scale greatly ex- 
panded near the resonance point of the crystal. At 
frequencies sufficiently removed from resonance, 
both above and below f„ where the motional im- 
pedance is large compared with Xc., the X, curve 
is essentially the same as the reactance curve of a 
capacitance equal to C,. X, is inductive only be- 
tween its two zero points, f, and f,. Note that R, 



Figure 1-100. Impedance characteristic* versus fre- 
quency ef crystal unit. Neither the frequency nor the 
impedances are drawn to scale 


WAOC TR 56-156 


80 



rises sharply to a maximum at f., where it is equal 
to the parallel-resonant impedance of the equiva- 
lent circuit of figure 1-97. Since R is much smaller 
than Xcx,, at antiresonance 

Re = Z. = (Xc„)VR 

In the case of the particular 10-mc crystal unit 
where Xco = —3.5 X 10*n, and R = 8fl, Re at 
antiresonance will be approximately 1.5 megohms. 
Ze(= VRe* 4- ■^’) most frequencies is simply 
equal to Xe. Only in the immediate regions of fr 
and f„ where X* becomes negligible, is the magni- 
tude of Ze affected greatly by Re. The impedances, 
of course, are not drawn to scale. For example, if 
Z, at antiresonance were drawn to the scale used 
for Ze at resonance, the curve could extend more 
than a mile above the horizontal axis. 

PARALLEL-RESONANT FREQUENCY, fp, 

OF CRYSTAL CIRCUIT 

1-210. Although an oscillator may depend upon a 
crystal operating at its series-resonant frequency, 
it is not practicable for a crystal unit to control 
an oscillator at the antiresonant frequency, f,. The 
crystal will either be operated to pass a maximum 
current (series-resonant circuit), or to develop a 
maximum voltage (parallel-resonant circuit) at 
some proper phase and frequency. It would seem 
that these latter conditions could best be met by 
operating the crystal unit at its antiresonant fre- 
quency, for it is in this region that the effective 
impedance is most sensitive to small changes in 
the frequency. However, another circuit, such as 
the input of a vacuum tube, will necessarily be 



Figure 1-101. Equivalent parallel-resonant circuit of 
crystal unit (X^p, R,p) shunted by load (X^., KJ. Nor- 
mally fr < fp < far M that X,p IS Inductlve and Xj. 
is capacitive 


Section I | 

Eloetrical Parameters of Crystal Units f 

shunted across the crystal. The shunt, or load cir- 
cuit into which the crystal operates will have a 
much lower impedance than that of the crystal 
at antiresonance, so that the total impedance will 
be relatively insensitive to small frequency varia- 
tions in the region of f.. In determining the actual 
frequency stability, the entire circuit must be con- 
sidered as a whole. The operating frequency may 
be considered as the resonant frequency, fp, of an 
equivalent parallel circuit, as shown in figure 
1-101. Xe„ and R„„ are simply the reactance and 
resistance of the equivalent circuit of the crystal 
unit at fp, and X^ and R, are the equivalent shunt 
reactance and resistance, respectively. Since X, is 
more frequency-sensitive above series resonance 
than below, there is normally no advantage in 
using a crystal in circuits that require X, to be 
capacitive. Thus, in practice, f,„ will be some inter- 
mediate frequency between f, and f., so that X^p is 
always inductive and X, is always capacitive. The 
distinction made between “parallel resonance” and 
“antiresonance” in this discussion is somewhat 
arbitrary, and it is not uncommon to use the term 
“antiresonant” to describe any parallel-resonant 
crystal unit. 

Effects of Changes in Shunt Capacitance on fp 

1-211. In discussing af,, the difference between 
the motional and the effective resonant frequency, 
it was found that 

That this quantity is normally insignificant is for- 
tunate, for it varies directly with the square of 
R, a parameter quite likely to change during opera- 
tion. On the other hand, it was later found that 



could amount to more than 0.1 percent difference 
in frequency. In this case, since, Af, is relatively 
large, it is also quite fortunate that, to a first 
approximation, the antiresonant frequency of a 
given crystal unit is independent of operational 
changes in R. However, it is not the antiresonant 
frequency of the crystal unit itself, but rather, 
the actual parallel-resonant frequency at which 
the crystal unit will operate that is of primary 
interest. Let fp — f, = Af„. Now, it can be imagined 
that Afp is simply the Af, of a crystal unit whose 
shunt capacitance C„ has been increased by an 


WADC TR 56-156 


81 




Section I 

Electrical Parameters of Crystal Units 






2'Vt 

W. C,* 

Ct* 


» Xtp 


Ct* Cp+C, 

V S V 2 

*»P *tp 

“ R+ Rj ’ R+Ry 


Figure I- 1 02. equivalent parallel-resonant tank cir- 
cuit, in which the motional impedance of the crystal 
unit is the inductive arm of the tank, and the total 
shunt impedance is the capacitive arm 

amount Ci and which has an effective resistance 

added to the shunt arm equal to Rj This 

last assumption can be made without introduc- 
ing an appreciable error as long as Rj is small 
compared with X,. The multiplying factor is 

needed, since only a fraction, f ^ ^ g 

total equivalent tank current will flow through R,. 
The equivalent tank circuit is shown in figure 
1-102, where Ct and Rt are the values of the shunt 
parameters. Now, since Afp is equivalent to the 


Af, of a crystal unit that has C„ = Ct, Afp will be 
expressed by the same general formula that 


holds for Af,. Thus, Afp = 



. Also, since 


Xt =— l/2TfCT, it can be seen that Af,, will be 
inversely proportional to the total shunt capaci- 
tance. Although C„, itself, is the most stable of all 
the crystal parameters, the stability of the effec- 
tive external capacitance C, will depend upon the 
over-all design of the oscillator circuit. The crystal 
unit may be considered a device that determines 
the limits within which the frequency may be 
varied; that is, fp must lie somewhere between f, 
and f.. However, it is primarily the parameters of 
the external circuit in conjunction with the equiva- 
lent L, C, and Co of the crystal that fix the exact 
frequency ; and although the stability of the crys- 
tal parameters is fundamentally a problem for the 
crystal manufacturer, the stability of the effective 
Ct is largely the concern of the radio designer. 


1-212. Figure 1-103 (A) shows the reactance curve 
of X, versus frequency, and figure 1-103 (B) shows 
the reactance curve of Xt versus Ct. The values 
of X, are those of the 10-mc crystal unit which 
has previously been taken as an example, and 
where L is assumed to be 1.65 X 10“’ henry. Note 
that the variations in Xt with frequency have been 
neglected, and f is simply assumed to equal the 
nominal frequency of 10 me, insofar as the capaci- 
tive arm is concerned. Since X, and Xt are drawn 
to the same scale, a horizontal line drawn through 



01 23 4 S 67e9IOIIl2l3t4ISICr7IS 4 68l0l2l4l6l8 20 22 24 2«2e 30 32 34 3e3a40 

Af (ke/KK) Crluut) 


WADC TR 56-156 


Figure 1-103. Reactance curves of: (A) X, versus Af,(B) —X, versus Ct 

82 




both curvea will intersect points of equal but oppo- 
site reactances. These points of intersection will, 
in turn, indicate the value of af required for a 
given Ct, if the two arms are to be resonant. For 
example, at Ct = 7.69 /i/if, Xt = —2000 n ; so that 
in order for X, to be 2000 n, afp must equal 9.65 kc. 
Likewise, a Ct of 40 /i/»f will mean approximately 
a afp of 2 kc. Now it so happens that the part of 
Ct represented by C, will have a component that 
tends to vary with changes in the plate voltage 
applied to the vacuum tube, changes in the tem- 
perature or the tuning, changes in the coupling 
and neutralizing adjustments, and any changes in 
the vacuum-tube characteristics or other circuit 
parameters due to other causes. Such a change in 
Xi will cause not only a change in the resonant 
frequency, but also a change in the amplitude of 
the oscillations. If a given change in Ci is to have 
a minimum effect upon the frequency and power 
expenditure of the oscillator, then Cr must be as 
large as possible without seriously reducing the 
stabilizing effect of the crystal. In other words, 
Ct should have a value where the slope of the 
Xt-vs-Ct curve is not steep. For the 10-mc crystal 
of figure 1-103, maximum stability would be ob- 
tained with Ct between 36 and 40 /n^f. With C„ 
equal to 4.55 nid, this would mean a load capaci- 
tance, Ci, between 32 and 36 /i.#if. As much of Ci 
as is possible should be supplied by a fixed or ad- 
justable capacitor connected directly across the 
crystal unit or in some other part of the circuit, 
so that its effective capacitance with respect to the 
crystal terminals will remain constant, and not be 
affected by changes in the tube characteristics. 
This would reduce the variable part of C, to a 
minimum. Cr, however, should not be made so 
large that Xt will approach the magnitude of R, 
otherwise the crystal will not only lose some of 
its stabilizing effectiveness, but will require an 
excessive drive level to maintain oscillations. 

Sfabilbin<| Effect of Crystal on fp 

1-213. Although Ct. plays an important role in the 
final determination of the frequency, it is the crys- 
tal itself that must be primarily responsible for the 
stability of the frequency — that is, if the use of a 
crystal is to be justified. For this reason, care 
should be taken to mdke certain that the apparent 

Q. of the crystal series arm as large as 

10, if possible, and preferably much larger during 
operation. Otherwise, the series-arm impedance 
will not respond with maximum sensitivity to 
changes in Ct. However, since Xt, and hence X„ 
must be kept small to reduce the effects of a 


Section I 

Electrical Parameters of Crystal Units 

change in Ct, it might appear at first thought that 
a conventional coil could serve quite as well as a 
crystal. The reason why this is not true is that 
the frequency stability is dependent upon the mag- 
nitude of the change in reactance for a given 
change in frequency, and not primarily upon the 
total magnitude of the reactance. It will be re- 
called that, in the conventional L-C circuit, the in- 
stantaneous rate of change of Xl with frequency is 



and that, at resonance 

dXL ^ 
df df 

In the parallel-resonant crystal circuit, however, 
these equalities do not hold, for X„ = 4*rafL, and 

not 2>rfL. Thus, = 4irL, where L of the crystal 

is greater than L, of a coil of the same reactance 
by a factor of f/2sf. Since at resonance, the rate 
of change of Xt would equal that of Xi,, it follows 
that 4irL, the change in the motional reactance 
with frequency, will be fp/Afp times as great as 
the change in Xt with frequency. Consequently, 
the stabilizing effect of the crystal is much greater 
than that of the shunt reactance, so that, for ail 
practical purposes, the crystal can “automatically” 
annul the effect of small changes in Ct, but not 
vice versa. It can be seen that, even with X, rela- 
tively small, the stabilizing effect of the crystal 
for a fixed change in Xt is not diminished, pro- 
vided, of course, that X, is sufficiently large, as 
compared with R, so that the total impedance of 
the series arm is essentially equal to, and varies 
linearly with, X„. (See paragraphs 1-238 to 1-245.) 

Effect on Parallel Crystal Circuit 
Due to Variations in Resistance 

1-214. As long as the apparent Q of the parallel- 

resonant circuit ^ ^ at least as great as 

10, a change in either R or Rt will not, in itself, 
have a large effect upon f„. However, depending 
upon the design of the particular circuit, a change 
in the resistance may indirectly affect the fre- 
quency by causing a change in Cr, since, to a cer- 
tain extent, the effective Ct will be a function of 
the other circuit parameters. The most critical ef- 
fect due to changes in the resistance parameters 
is the effect on the power required for excitation 
of the oscillator in order to obtain a given output. 
The impedance, Zp, of the parallel circuit at reso- 


WADC TR 56-156 


83 



Section I 

Etectricol Parameters of Crystal Units 

X * 

nance will be approximately p .‘ ‘ *p . An increase 

Jv — p iVx 

in the total resistance of 100 percent would thus 
decrease Zp by one half. If , for example, the output 
of the oscillator depended directly upon the r-f 
voltage across Zp (i.e., across the crystal), a de- 
crease in Zp by one half would require twice as 
much power in the crystal circuit to maintain the 
output at the same level as before. A part of Rt 
will be the result of reflected resistance losses in 
the output circuit. An increase in the load will thus 
be reflected as an increase in Rt. This is unfortu- 
nate, for if the load should increase it would be 
desirable to have an increase in Zp, to raise the 
excitation voltage automatically, or at least to keep 
it constant. As it is, the effect is to decrease the 
excitation, unless special circuits, such as the Tri- 
Tet, are employed to increase the feedback directly. 
If a principal component of the losses in Rt are due 
to the losses in the grid circuit, and if the oscilla- 
tor design is such that the grid current is not 
linear with the excitation voltage, but rises at a 
much greater rate, then Rt can rapidly increase 
or decrease with the excitation voltage, and Zp will 
vary inversely. Under these conditions, Zp will 
always change in a direction that will tend to annul 
any change in the excitation voltage. The greater 
that part of Rt reflecting the grid losses, as com- 
pared with that part reflecting the output losses, 
the greater will be the amplitude stabilizing effect 
for counteracting changes in the plate voltage or 
the effective load resistances. Another character- 
istic of a crystal circuit in which Rt varies auto- 
matically is that the effect resulting from a varia- 
tion in R is minimized. Assume, for example, that 
the desired output at a constant load will require 
a certain effective value of Zp. If, for some reason, 
R should change, thereby changing the excitation 
voltage, Rt would tend to change by an equivalent 
amount in the opposite direction, thus maintain- 
ing Zp, and hence the output, essentially constant. 
However, a change in R or Rt will almost certainly 
be accompanied by a change in the crystal power 
losses, thereby causing a frequency drift if the 
particular crystal unit is frequency-sensitive to 
the drive level. At this point, however, the impor- 
tant items to note are: (1) Xt,, and hence X„ 
preferably should not be smaller than IO(R-I-Rt), 
or the maximum stabilizing effect of the crystal 
will not be realized; (2) the direct effect of a 
change in (R -j- Rt) is to change Zp; (3) the ef- 
fects of a change in Zp primarily will involve 
changes in the excitation voltage, in the power 
expended in the crystal circuit, as well as that de- 
livered to the load, and in the equivalent value of 


Ct, thereby also changing the frequency; (4) if 
changes in either R or Rt are such that the power 
expended in the crystal unit itself is caused to 
vary, then a significant change in the frequency 
characteristics of the crystal may result; and (5) 
for maximum frequency stability, the oscillator 
should be lightly loaded, and the drive level of the 
crystal should be as small as is practicable. 

Effect on Parallel Crystal Circuits Due to 
Variations in Motional-Arm C or L 

1-215. Crystal circuits operated at the resonant 
frequency of the crystal units may be only slightly 
affected by variations in C or L from one crystal 
unit to the next, or even during the operation of 
a particular unit, provided that the effective LC 
product remains constant, so that the frequency 
does not change. In the parallel-resonant circuit, 
however, even if f, is the same, a different C and L 
means a change in Afp. For a given Ct and nominal 
frequency, Xtp, and hence, X.p, must remain ap- 
proximately constant, so that Afp, equal to X,p/4irL, 
will tend to vary inversely with L. The exact value 
of L for a given crystal unit will depend upon the 
effective electrode area, the orientation of the cut, 
the thickness of the crystal, whether twinning is 
present in the quartz, and the degree to which 
spurious modes are coupled to the desired mode. 
Insofar, as the variations in L from one crystal 
unit to the next are concerned, no problem arises 
unless it is necessary to adjust fp to an exact value ; 
in which case the problem of the design engineer is 
to ensure that Ct will be sufficiently adjustable so 
that the desired fp may be obtained with any 
reasonable value of L. Since such adjustments 
must be provided for anyway, in order to allow for 
different values of f„ no new problems are intro- 
duced, except that a greater deviation in Afp must 
be met than otherwise. Unless spurious modes are 
closely coupled to the desired mode, the variations 
in L that might oc^ur during the operation of a 
particular crystal unit will be too small to affect 
the magnitude of Afp, as long as Xtp remains con- 
stant. However, the operational variations of L 
and/or C may be such that f, will change, in which 
case fp will also change. Such a deviation in fre- 
quency, i.e., in the equivalent LC product, would 
occur during changes in temperature or drive level, 
or because of fatigue or other aging effects. Mini- 
mum variations in L and C are obtained by the 
use of low temperature-coefficient crystals and 
constant-temperature ovens, and by ensuring that 
the drive level will remain both low and constant. 
In any event, a reasonable operational variation in 
fp can be compensated for by an adjustment in Ct- 


WAOC TR 56-156 


84 



MialmHin Value of Afp 

1-216. Returnin^r to equation 1 in figure 1-98, let 
it be imagined that X, represents the effective re- 
actance of the motional arm of a crystal unit in 
parallel with a total capacitance Cr, instead of 
simply the C« of the crystal unit, itself. Further- 
more, assume that Rt is negligible. As before, the 
condition of resonance is that X* be zero, which 
will occur only when 


R" -1- X/ -f- X. Xt = 0 


(Note that Xt now replaces X,;„.) Now X, = 4irLAf, 
and on substitution in the preceding equation and 
rearranging, it is found that 


(Af)' 


Xt 


^ =0 


Note that this is simply a quadratic equation of 
the type AX’ -|- BX C = 0, so by the quadratic 
formula 


Xt /Xt' - 4R^ 

4tL \ l&r'L' 


The ± term indicates that there are two possible 
solutions for Af at which resonance will occur. One 
of these is equivalent to Af, (but with Co replaced 
by Ct)» and the other is equivalent to Afp. For 
these solutions of Af to be real, Xt’ must be greater 
than 4R’; otherwise, the expression under the 
radical sign becomes negative, and Af will be 
imaginary. However, in the special case where 
Xt’ — 4R’ = 0, there is only one solution for Af. 
In other words. 


Af, 


Afp 


-Xt 

8irL 


This represents the minimum value obtainable for 
Afp ; or, from the point of view of series resonance, 
it may be considered the maximum value obtain- 
able for Af,. The important point to note is that 
neither parallel nor series resonance is possible 
unless Xt* is equal to, or greater than, 4R’. At the 
minimum Afp, 


Xt* = 4R' 
or 

I Xt/R i = X./R = 2 

It should be remembered that all the resistance 
has been assumed to be in the motional arm, and so 


Section I 

Electrical Parameters of Crystal Units 

has the effect of limiting the maximum component 
of lagging current for a given voltage ; but parallel 
resonance could be achieved at any frequency be- 
tween f. and f, if Rt were equal to R. However, 
since C„ limits the minimum amplitude of leading 
current, Rt cannot be made equal to R for all 
values of fp, and there will still be a minimum fp 
greater than f,. (A minimum which can be shown 
to be identical with the natural f, of the crystal 
unit.) With Rt assumed to be negligible, the ratio 
of reactance to resistance equal to 2 represents 
the minimum apparent Q. of the parallel crystal 
circuit, if resonance is to be obtained. As stated 
previously, if the full stabilizing properties of the 
crystal are to be in use, Q, should be at least 10. 
However, if the power delivered to the crystal cir- 
cuit is sufficient, oscillations can be maintained as 
long as the apparent Q. does not fall below 2. This 
occurs at the frequency at which the amplitude 
of the lagging component of current through the 
series arm is the maximum obtainable. 

TYPICAL OPERATING CHARACTERISTICS 
OF CRYSTAL UNIT 

1-217. Figure 1-104 shows the effective impedance 
characteristics of the 10-mc crystal unit which has 
been assumed to have the following parameters: 
L = 1.65 X 10“* henry 
C = 1.54 X 10-’ /x/*f 
R = 8 ohms 
Co = 4.55 n/if 

Xe, R„ and are given by equations 1, 2, and 3, 
respectively, in figure 1-98; X(:„ is assumed to be 
equal to —3.5 x 10* ohms for all values of Af. Note 
that the normal operating range covers only about 
one fourth of the total range between f, and f,. 
Of course, if Co were greater than the value as- 
sumed, Af, would be smaller and the normal opera- 
ting range would be a larger percentage of the 
total. As explained previously, Ct ( = Co + C,) 
must be relatively large, so that small variations 
in C, will not greatly affect the frequency, and it 
is this consideration that limits the practical oper- 
ating range to low values of Af. At parallel reso- 
nance, Xe must approximately equal — X,, for the 
same reason that X, must equal — Xt. Standard 
military high-frequency crystal units are normally 
tested with a value of C, = 32 ii/if. At 10 me, a 
capacitance of 32 /i/,f will have a reactance of 
approximately — 500fJ, as indicated in figure 
1-103 (B). With Co = 4.55 ^^f, Ct will be 36.55 ^id, 
which corresponds to a value of Xt = —440 O, and 
a Af = 2.1 kc/sec. Af can also be found from the 
reactance curve of figure 1-104 at the point where 


WAOC TR 56-156 


85 



Section I 

Electrical Parameter* of Crystal Unit* 



O I 2 5 4 S 7 e 9 10 II 12 13 14 IS 16 17 18 19 

Af IN KC/SEC 


Figure 1-104. Typical characterittic curvet for X, and R, at 10-tttc crystal unit. 
(Shunt resistance, R/, across crystal is assumed to be negligible) 


Xe = — Xi = 500 fi. Crystal units are sometimes 
operated in series with an external capacitor, C,, 
as indicated in figure 1-105. Slight variations in 
the frequency can be compensated for by adjust- 
ments of Ci, and resonance will occur at the fre- 
quency at which X, is exactly annulled by X*. If 
the ratio of Xe/Re is sufficiently large, then for all 
practical purposes the series-resonant frequency, 
f„, is the same as the fp of the crystal unit in 
parallel with the same C,. At resonance, the crystal 
unit and Ci in series have an effective impedance 
equal to Re- Although there is an effective maxi- 
X Af 

mum Qpm = ^ when Af = , it has no special 

significance directly concerning the frequency sta- 



Figure 1-105. equivalent circuit of crystal unit 
connected in series with capacitor 


bility of the circuit, but does tend to increase the 
activity. In general, if a series capacitor is used, 
its reactance will be small, as compared with Xc«. 
Indeed, it may be used for no other purpose than 
to annul the self inductance of the crystal leads. 
It can be seen in figure 1-104 that R* does not in- 
crease nearly as rapidly as does Xe, except in the 
region of f,. With Ci = 32 /i/if, Af,, (= f„ — f,) 
will be 2.1 kc for the crystal unit of figure 1-104, 
and R, will be between 11 and 12 ohms. 

MEASUREMENT OF CRYSTAL PARAMETERS 

1-218. The parameters L, C, R, and C„ of any crys- 
tal unit chosen at random are effectively four inde- 
pendent variables, so that a minimum of four 
measurements are required to determine the 
values of these variables. Probably the four easiest 
measurements to make are those for f„ R, Co, and 
f„. The measurement for the last quantity is made 
when a known load capacitance C, is connected in 
series with the crystal unit. Since fn, the resonant 
frequency of the crystal and Cj in series, for all 
practical purposes will be equal to the fp of the 
crystal in parallel with Ci, we shall normally not 
make a distinction between the two frequencies in 
the following discussion, but shall use the symbol 
“fp” in referring to either. 


WADC TR 56-156 


«6 


M*asiir«iii*at of tha Shunt Capucitaneu, C, 

1-219. At all frequencies sufficiently removed from 
resonance the crystal unit will have the character- 
istics of a capacitance equal to C,. Thus, at these 
off-resonance frequencies, Co can be measured by 
a conventional Q meter or an r-f bridge. The fre- 
quency at which C. is to be measured should be 
lower than, but reasonably close to, the operating 
frequency, particularly if the crystal unit is to be 
operated at a high harmonic mode in the v-h-f 
range. Otherwise, the effect of the distributed in- 
ductance of the leads will not be properly taken 
into account. 


Muaturemeat of the Series-Arm Resistaace, R 

1-220. R is normally measured with the aid of a 
Cl meter (crystal impedance meter). (See also 
paragraphs 2-60 through 2-6.5.) There are four 
standard Cl meters with which the crystal units 
described in Section II of this handbook have 


been tested : 

Crystal Impedance , 
Meter 

TS-710/TSM 

•TS-537/TSM 

TS-330/TSM 

TS-683/TSM 


Frequency Range 
(kc/sec) 

10 to 1100 
75 to 1100 
1000 to 15,000 
10,000 to 75,000 


• (Crystal Impedance Meter TS-537/TSM may soon 
be replaced entirely by the recently developed 


Section i 

Electrical Parameter* of Cryctal Units 

Crystal Impedance Meter TS-710 ( )/TSM.) A Cl 
meter is essentially an r-f oscillator provided with 
a feedback circuit in which a crystal unit or a re- 
sistor, or a crystal unit in series with a calibrated 
capacitor, can be connected. A simplified schematic 
diagram of a typical Cl meter is shown in figure 
1-106. The circuit shown is a modified Colpitts 
oscillator in which the tank inductor has been ef- 
fectively divided into two equal sections, L, and 
Lj, between which a resistor, R„, equal to R„ the 
effective resistance of the crystal unit, can be con- 
nected. The ganged tuning capacitors C, and Ci 
are at all times equal. Ch is simply a blocking 
capacitor to isolate the plate voltage from the crys- 
tal terminals, and adds only a negligible reactance 
to the tuned circuit. The potentiometer, P, is used 
to control the screen grid voltage, and hence the 
r-f output of the tube and the drive level of the 
crystal. With S, connected as shown, and R„ ad- 
justed to a value typical of the motional-rrm R for 
the type of crystal unit being measured, the circuit 
will oscillate at approximately the resonant fre- 
quency of the tank. If C, and Cj are adjusted so 
that the natural frequency of the oscillator is near 
the nominal frequency of the crystal, then, on con- 
necting the crystal into the circuit, oscillations 
will continue, but with the frequency determined 






WADC TR 56-156 


87 





Section I 

Electrical Parameters of Crystal Units 

by the total reactance of the tank, including X, 
of the crystal unit. There is no standard practice 
as to grounding the crystal holder; but whether 
grounded or ungrounded, the method of connect- 
ing the crystal unit should be noted. The drive is 
adjusted so that a very small grid current is indi- 
cated on M,. Under these conditions the control 
grid is positive with respect to the cathode only 
at the peaks of the positive swings of the excita- 
tion voltage developed across C,. Since electrons 
flow from cathode to grid only at these instants, 
a small percentage change in the excitation volt- 
age, as illustrated in figure 1-107 can cause a very 
large percentage change in the cathode-to-grid 
electron flow. In an actual circuit the idealized 
constant bias that is indicated in figure 1-107 does 
not occur because of the gridleak action. However, 
if the gridleak contribution to the bias is very 
small compared with that part developed across 
the cathode resistance, the increase in bias due to 
an increase in excitation occurs almost entirely 
across R*, not across the cathode resistance. 
Hence, a five or ten per cent increase in the total 
bias can result from hundred per cent increase in 
the gridleak IR drop. In this way, the grid current 
meter is a very sensitive indicator of slight 
changes in the r-f voltage across C„ and hence of 
any change in the tank current. With S, in the 
crystal position and S 2 closed, as C, and C, are 
varied, a peak in the grid-current reading indicates 


a maximum current through C,. This in turn 
means that the effective resistance of the crystal 
unit has reached the minimum value equal to the 
series-arm R. In other words, the oscillator fre- 
quency is coinciding with the resonant frequency, 
fr, of the crystal unit. R„ can now replace the crys- 
tal in the circuit and be adjusted to give the same 
meter readings at the same frequency. At this 
point Ro will equal R, and, since R„ is known, R 
will have been measured. The crystal current 
meter, M 2 , is not sufficiently sensitive to permit an 
accurate observation of the small changes in tank 
current that occur as the circuit is tuned through 
f,. The purpose of the meter is to decrease the 
possibility of overloading the crystal and to pro- 
vide a ready me&ns for determining the exact drive 
level at which the crystal is being tested. Since the 
crystal parameters may change with the drive, it 
is necessary to specify the drive level at which the 
measurements are made. Expressed in milliwatts, 
*the drive level equals PR X 10~S where I (in 
milliamperes) is the current through Mj at f,. If 
a crystal-current meter is not supplied, two vac- 
uum-tube voltmeters can be used to measure the 
voltage from each crystal terminal to ground with- 
out seriously affecting the circuit. Where E is the 
difference in potential across the crystal, equal to 
the difference between the two terminal voltages, 
the drive level is equal to EVR- The temperature 
at which the measurements are made should also 


o 

UJ 

a. 

CO 


o o 

OEO 
o X 

-J < 

o o 
f o 

Z I- 

o 

o 


o 

lU 

o 

< 


o 

> 



LJ 

q: 

a: 

D 

o 


a: 

o 


Figure 7-707. How a small percenfage change in excitation voltage can cause a large percentage change in 
grid-leak current of Cl meter. Bias does not actually remain constant as indicatedf but follows the percentage 
changes of excifafion voltage. Nevertheless, the relative percentage variations of grid current and excitation 
voltage can be approximately as shown when the greater part of the bias 
is developed across the cathode resistance 


WADC TR 56-156 


86 




be specified, although, in general, the variation of 
R with ambient temperature is much less than 
its variation with the amplitude of the crystal 
vibrations. 

Measurement of the Resonance Frequency, f, 

1-221. To measure f,, a c-w radio receiver, a radio- 
frequency standard, a calibrated audio-frequency 
source (interpolation oscillator), and either a 
loudspeaker, a pair of ear phones, or an oscillo- 
scope are used in conjunction with the Cl meter. 
With the crystal connected in the Cl-meter circuit 
and the oscillator tuned to series resonance, the 
r-f output can be loosely coupled through a coaxial 
cable to the antenna post of the c-w radio receiver. 
After the receiver is tuned to the frequency of the 
crystal unit, the Cl meter is turned off, and the 
receiver is connected and tuned to receive the par- 
ticular harmonic of the frequency standard that 
is nearest to f,. The bfo of the receiver is then cut 
off, and the Cl meter is turned on. With both the 
standard and the Cl-meter signals being fed to 
the receiver input, the output of the receiver will 
be an audio beat note equal to the difference be- 
tween the known standard frequency and the un- 
known crystal frequency. By momentarily switch- 
ing a fairly large value of C, in series with the 
crystal, so that the Cl-meter frequency increases 
slightly, the audio beat frequency will rise or fall 
according to whether f, is respectively greater 
than or less than the standard signal. The audio 
beat frequency is next mixed with the audio out- 
put of the interpolation oscillator, which in turn 
is adjusted to bring the beat frequency of the two 
audio signals to zero — the zero beat being ob- 
served by phones, loudspeaker, or oscilloscope. At 
zero beat, the crystal frequency will have been 
measured to be equal to the selected r-f standard 
frequency ± the interpolation oscillator frequency. 
The accuracy of the measurement depends pri- 
marily upon the accuracy of the frequency stand- 
ard, and secondarily on that of the interpolation 
oscillator. As in the case of the resistance meas- 
urement, both the temperature and the drive level 
should be specified, and these should be the same 
as when the measurement of R was made. 

Measurement of the Parallel-Resonance 
Frequency, f,, 

1-222. To measure f,„ it is first necessary to adjust 
the Cl meter to oscillate at f„ by the same steps 
employed previously. With the oscillator so ad- 
justed, a known value of is switched in series 
with the crystal. The new frequency will be ap- 
proximately equal to fp. The more common values 


Section I 

Electrical Parameters of Crystal Units 

of Cl are 32 /i./if for high-frequency crystals, and 
20 ([t/if for low-frequency crystals. In testing to 
determine whether the crystal frequency is above 
or below the frequency of the test standard, it may 
be more convenient to add capacitance across the 
crystal unit than to change the setting of C^. In 
this event the Cl-meter frequency is decreased 
rather than increased, so that the effect on the 
beat note will be the opposite of that previously 
described. Simply touching the crystal holder with 
the hand is normally the quickest method of in- 
creasing the shunt capacitance; however, care 
should be taken that the method employed does 
not effectively decrease the capacitance by ground- 
ing the holder. 

1-223. Theoretically, the foregoing method of 
measuring f„ is not exact, for if the LC circuit is 
correctly tuned when the crystal appears as a pure 
resistance, the same feed-back phase relations can- 
not hold at the higher frequency, f,„ unless CR 
and C, in series introduce a negative reactance to 
compensate for the increase in Xj,, and Xlj and 
the decrease in Xc, and X^j. In other words, X. of 
the crystal unit is approximately, but not exactly, 
equal in magmitude to X, of the load capacitance. 
Actually, Xe is less than X,| by an amount ap- 
proximately equal to the change in reactance 
around the LC loop, exclusive of CR and C,. This 
change is approximately equal to 4jr(L, -|- L,) Afp, 
where Af,, is the difference between f„ and f,. At 
the true fn, X, -f X, = 0; at the observed fp, 

Xe -h X, -H 4ir (Li L 2 ) Afp = 0 

1—223 (1) 

Now a small change in X,. -j- X^, equal to aX* -4- 
aXx, as a result of a small change in frequency, 
is practically equal to AXe alone. If the frequency 
is sufficiently close to the resonant point, f„ we 
may set X, -f X^ (at observed f,,) = A(Xe -f- X,) 
== aX, <== aX, = 4-rLAf, where Af = observed fp 
— true fri. By substitution in equation (1) 

True == Observed fp + ^ ^ ^ ^ Afp 

1—223 (2) 

Maasaramanf of the Effective Resistance, R„ 
at Parallel Resonance 

1-224. In the measurement of fp, the drive level 
and the temperature should be the same as in the 
measurement of R and f,. To determine the drive 
level, either the voltage across the crystal unit, or 


WADC TR 56-156 


89 



Section I 

Electrical Parameters of Crystal Units 

the effective resistance should be known — or 
both, if a crystal-current meter is not provided. A 
measurement of Re is also important for its own 
sake and as a check to see whether the motional- 
arm parameters are the same at fp as at f,. In the 
case of a crystal unit which is intended to be oper- 
ated only at parallel resonance, R, is generally 
treated as a primary parameter of more immediate 
importance than the motional-arm R. R, is meas- 
ured in a manner similar to the measurement of 
R, except that on substituting R„ for the crystal 
the circuit must be retuned so that oscillations are 
being maintained at fp. For a very precise drive- 
level measurement, additional precautions must be 
taken if the power dissipation is to be the same 
■ n both the series- and parallel-resonant measure- 
ments. The best assurance that the f, and fp drive 
levels will not be greatly different is to be had 
when the crystal current is kept near the minimum 
necessary to maintain oscillations. Thus, even 
though the relative differences in drive level may 
be large, the absolute differences will be small. This 
is not a completely reliable method, for some crys- 
tal units exhibit very sharp increases in resistance 
when the drive level approaches a minimum. 

Computing the Series-Arm C and L 
from the Measured Parameters 

1-225. From the formulas for f,, Xp, and Xi, it is 
quite easy to derive the following approximate 
equations for the series-arm parameters, C and L : 


C = 


2(Co -f CJAfp 

fr 


(27rf,)^C 


1—225 (1) 

1—225 (2) 


where, C, C„, and Cx are in farads, L is in henries, 
and Afp and f, are in cps. 


METHODS FOR EXPRESSING THE RELATIVE 
PERFORMANCE CHARACTERISTICS 
OF A CRYSTAL UNIT 

1-226. If the four equivalent electrical parameters 
(L, C, R, C„) are accurately known for a given 
state of operation, no other independent data con- 
cerning a crystal unit can increase the radio engi- 
neer’s knowledge of how the crystal will perform 
under the given conditions. However, the radio en- 
gineer has been slow in requesting specific infor- 
mation concerning the electrical characteristics of 
the crystal units available, and as a result the prob- 
lem of making a given circuit perform correctly 


has often in the past effectively become the re- 
sponsibility of the crystal manufacturer, who, by 
cut-and-try methods, has been more or less 
required to design the crystal unit around the 
particular circuit. Fortunately, progress toward 
greater standardization of crystal units has been 
considerably accelerated during recent years be- 
cause of the increased demands of the military 
services ; but there is still a tendency on the part of 
the design engineer to regard a crystal unit, as 
one production engineer has expressed it, as a 
“mystery box," rather than the equivalent circuit 
that it is. Contributing to this tendency has been 
a hesitancy upon the part of the manufacturer to 
describe his crystal units in terms of the most 
probable equivalent electrical parameters. At the 
present state of the art, wide variations from the 
most probable values can occur, and the manufac- 
turer quite naturally wishes to avoid the chance 
that typical values of the parameters will be mis- 
interpreted as specified values. For similar rea- 
sons, a description of a crystal unit in terms of its 
most probable parameters is not at present desira- 
ble from the point of view of the military services, 
lest a crystal circuit be designed upon the assump- 
tion that the typical crystal parameters will always 
be available, rather than upon the assumption that 
the crystal unit cannot be depended upon to meet 
other than its minimum performance specifica- 
tions. If the former, rather than the latter assump- 
tion were made, a carefully designed circuit might 
fail to operate properly if used with a borderline 
crystal unit. Thus, the purpose of the standardiza- 
tion of types — to ensure a complete interchange- 
ability among the crystal units of the same type 
number and nominal frequency — would be de- 
feated. Nevertheless, the lack of emphasis upon 
the basic parameters has served to cloak the crys- 
tal in an air of mystery, and to instill in the radio 
engineer an impression that a crystal circuit is 
possessed of properties that cannot be expressed 
in the normal idiom of LCR networks. Contribut- 
ing somewhat to this point of view is the special 
terminology that has been developed for the pur- 
pose of comparing the performance characteris- 
tics of one crystal unit with those of another par- 
ticularly where the definitions of the terms contain 
certain ambiguities or conditional interpretations, 
or are presented as mathematical relationships 
without concrete qualitative meanings. What may 
be implied as a property of the crystal unit alone, 
may well be a function of the particular circuit in 
which the crystal unit is mounted. Much of the 
difficulty can be avoided if it is kept in mind that a 
crystal unit has no important circuit performance 


WADC TR 56-156 


90 


qualities that cannot be expressed in the everyday 
terminology of radio engineering as it might apply 
if the crystal unit were replaced by an equivalent 
network of L, C, R, and C». 

1-227. There are five general categories in which 
crystal units can be placed for comparison insofar 
as their relative merits are reflected by their per- 
formance in a standard test-oscillator circuit : ( 1 ) 
activity, (2) frequency stabilization, (3) band- 
width, (4) quality factor, and (5) parameter sta- 
bility. Activity, as applied to a crystal, is a general 
term, rather loosely defined, that refers to the 
relative ease with which a crystal may be caused 
to maintain oscillations. The basic parameter most 
closely associated with the crystal activity is the 
motional-arm resistance, R. Besides R, or R^, there 
are certain performance parameters that can be 
used as indices of relative activity quality. These 
are the effective Q (Q^), the maximum effective 
Q (Qcm), the figure of merit (M) , and the perform- 
ance index (PI). The term, frequency stabiliza- 
tion, as used in this context, refers only to the 
ability of a crystal to minimize any change in the 
frequency due to variations in the parameters of 
the external circuit, In this sense, those perform- 
ance parameters that can be used as indices of the 
frequency-stabilization quality are the series-arm 
L/C ratio, the coefficient of frequency stability 
(Fx), and the capacitance ratios Ct/C and Ct/Cx- 
The bandwidth of a crystal unit refers to the fre- 
quency range over which the crystal unit is con- 
sidered operable. The performance parameters 
indicating this quality are the capacitance ratio, 
r = Co/C, and the electromechanical coupling fac- 
tor, k. The quality factor is simply the crystal Q, 
which is, itself, a major performance parameter, 
but one that is not exclusively identified with any 
one of the other four performance categories. The 
term parameter stability is used here to refer to 
the relative stability of the crystal parameters 
during changes in the temperature, drive level, 
tuning adjustments, and the like. The frequency 
stability of the crystal unit, which is included in 
this category, should not be confused with the 
function of frequency stabilization which is the 
characteristic we have arbitrarily assigned to the 
second performance category. The frequency sta- 
bilization is dependent upon the magnitudes of the 
equivalent-circuit parameters; whereas, the fre- 
quency stability is dependent upon the stability of 
the equivalent-circuit parameters. The stability of 
a crystal oscillator circuit is dependent upon both 
the crystal stabilization and the parameter stabil- 
ity. Performance indices or terms indicating the 
relative parameter stabilities are represented by 


Section I 

Electrical Parameter* of Crystal Unit* 

the temperature coefficients of frequency and re- 
sistance, drive-level coefficients of frequency and 
resistance, frequency tolerance, frequency devia- 
tion, resistance deviation, relative freedom from 
unwanted modes, and general expressions indi- 
cating durability and aging characteristics. Since 
most of the characteristics identified with the five 
performance categories can be expressed as func- 
tions of the same basic equivalent-circuit param- 
eters, a performance parameter in one category 
quite often serves as an indication of the crystal 
quality in another. It cannot be said that those 
properties most closely identified with the activ- 
ity, foe instance, are not also related to the fre- 
quency stabilizing effect. Nevertheless, classifying 
the various methods for rating the performance 
of a crystal unit is helpful in interpreting the dif- 
ferent performance parameters in terms of the 
basic equivalent-circuit parameters. 

Activity Quality of Crystal Unit 

1-228. The “activity” of a crystal oscillator is a 
qualitative expression referring to the amplitude 
of the oscillations. It is a term that came into use 
during the early days of crystal resonators, but 
one that seems never to have been vigorously de- 
fined. For example, it is not always certain 
whether the “activity of an oscillator” is intended 
to refer to the amplitude of current in the feed- 
back, or in the output circuit, or to the voltage 
across some particular circuit component, or to 
the output power, or to the excitation power, or to 
the ratio of these powers, or simply to the ampli- 
tude of the crystal’s mechanical vibrations. Were 
the expression not already so strongly entrenched 
in the crystal terminology, its use would probably 
be discouraged. As it is, crystal units are com- 
monly described as having high or low activities, 
or more specifically, as having high or low poten- 
tial activities or activity qualities. It will be found 
that the crystal parameter most directly indica- 
tive of the activity quality is the motional-arm con- 
ductance, 1/R. In crystal oscillators employing 
gridleak bias, when one crystal is replaced by 
another of the same nominal frequency, one of the 
crystals is usually found to produce stronger ex- 
citations and hence a larger grid current under 
similar operating conditions. Frequently the rela- 
tive grid currents are defined to be equal to the 
relative activities of the crystals. With this method 
of measurement it can be seen that, if a crystal is 
connected directly across the grid-to-cathode input 
the excitation, and hence the activity, will depend 
upon the amplitude of the r-f voltage across the 
crystal. On the other hand, if the crystal is con- 


WADC TR 56-156 


91 



Section I 

Electrical Parameters of Crystal Units 

nected in series with the oscillator input, the activ- 
ity will depend upon the amplitude of the current 
through the crystal unit. Since, in any event, the 
grid current depends upon the values of every 
parameter in the oscillator circuit, such a meas- 
urement is ambiguous unless a standard test cir- 
cuit can be referred to for each frequency. Only 
in this way can the crystal unit, itself, be con- 
sidered the only significant variable. Even under 
the assumption of ideal test conditions, however, 
the exact mathematical relationship among the 
crystal parameters, which provides the most direct 
measure of a crystal unit’s inherent activity qual- 
ity, has been a subject of some controversy. A 
number of suggestions have been made, but the 
usefulness of each of these depends considerably 
upon the method by which the crystal is to be used 
to control oscillations. As the crystal terminology 
becomes more rigorously defined we can imagine 
that the word “activity” will fall into disuse even- 
tually, with “effective resonance resistance” or 
“conductance” taking its place. 

ACTIVITY QUALITY FOR SERIES 
RESONANCE 

1-229. As an example of series-mode operation, 
we refer to the test circuit in figure 1-106. It can 
be seen that the grid excitation will be approxi- 
mately equal to IXc, the r-f voltage developed 
across C,. X<:, depends upon the frequency and the 
value of C„ whereas, I, the current through Ci 
depends upon the B+ voltage, the setting of P, the 
tube characteristics, etc., as v.'ell as the tank im- 
pedance and hence the resistance of the crystal. 
With all the circuit parameters constant, the only 
variable that the crystal introduces at resonance 
is its resistance R. Rather than specify all the 
parameters of the test circuit for each nominal 
frequency, it is clear that the measurements of R 
provide sufficient indication of the relative activi- 
ties of different crystal units under any similar 
conditions of resonance. Since the current and 
voltage amplitudes vary inversely with R, the 
series-resonance activity of any crystal unit can 
be assumed to be directly proportional to 1/R, the 
motional-arm conductance. 

ACTIVITY QUALITY FOR PARALLEL 
RESONANCE 

1-230. The interpretation of the activity quality of 
a crystal unit becomes more complicated when the 
crystal is to be operated at parallel resonance. But 
even as in the case of series resonance, the inher- 
ent pi'operty of the crystal unit that most readily 
indicates the relative activity is the motional-arm 


conductance, 1/R. In a parallel-resonant oscillator 
circuit, the excitation is normally directly propor- 
tional to the voltage developed across the crystal 
in parallel with its effective load capacitance Ci. 
This voltage in turn is proportional to Zp XtVR. 
where Zp is the parallel-resonance impedance, Xt 
is the reactance of Ct, the total shunt capacitance, 
and R is the series-arm resistance. As long as Xt 
remains constant, the only significant crystal vari- 
able that affects the activity is R, or more directly 
1/R. A complication arises from the fact that 
Ct = Co -|- Ci, so that if Ci, the effective capacity 
of the external circuit is to be held constant, then 
Xt, and hence the activity, changes with C„. 
Another complication arises when a measure of 
crystal quality is desired that will hold between 
crystals of different nominal frequency. If Ct or 
Ci is to be held constant, then Zp tends to vary 
inversely with the square of the frequency. Fi- 
nally, the complications are multiplied several 
fold when one begins to take into account the many 
ways in which a crystal can be connected to sta- 
bilize or to control a parallel-resonant type of cir- 
cuit. Unless the term “activity” is to refer to some 
desired and well-defined end result that can be 
measured quantitatively, the word may have little 
practical meaning. With this in mind, we note that 
since the usefulness of the oscillator depends en- 
tirely upon its output, the useful oscillator activity 
can be said to concern only the amplitude of oscil- 
lations in the load circuit. Thus, the relative ease 
with which a crystal unit enables a given output to 
be achieved under specified conditions can be re- 
garded as the relative activity quality of the crys- 
tal unit. From this point of view, the relative 
activity of a crystal unit can be considered in- 
versely proportional to the driving power that the 
crystal requires in order to maintain a given out- 
put level in a fixed load. Now, there is a special 
case of parallel-mode operation — where the series 
arm operates into a constant Ct — for which the 
activity, as interpreted above, requires only a 
measurement of R. Assume that a small, variable 
shunt capacitance, Cy, can be connected directly 
across each crystal unit whose activity quality is 
being tested, so that the effective shunt capaci- 
tance (eff Co = Co -h Cy) can be adjusted to give 
the same value for all crystals. With this arrange- 
ment, the crystal power dissipation for a given 
power output will vary positively with the mo- 
tional-arm resistance, R. In other words, two crys- 
tal units of the same series-resonance frequency 
and the same motional-arm resistance can produce 
the same oscillator activity at the same parallel- 
resonance frequency regardless of the particular 



WADC TR 56-156 


92 



/ 

1 values of Co, or of the motional-arm C and L, but 
only if a variable shunt capacitor is provided by 
which the effective Co can be held constant. Thus, 
for gauging the potential activity of a particular 
crystal unit to be used in any parallel-mode oscil- 
lator circuit having a variable capacitor connected 
directly across the terminals of the crystal unit, 
the motional-arm conductance, 1/R, can usually 
be considered a . sufficient and proper activity 
parameter. 

1-231. Where the load capacitance, Ci, that the 
entire crystal unit faces (not necessarily Ct = 
Co -|- Cl that the series arm faces) is to remain 
constant, the parameter 1/R is not a sufficient 
index of the activity quality. If the proper meas- 
ure of crystal activity is defined to be the ratio of 
output power to crystal power, such a definition 
is general enough to be applicable for any type of 
crystal oscillator. Certainly, the crystal unit that 
requires the least expenditure of energy to per- 
form its task should be considered the one of great- 
est activity. Unfortunately, activity quality, from 
the point of view of a power ratio, becomes a 
function of each particular oscillator and load cir- 
cuit, so that the generalization gained in the defi- 
nition is completely lost on application, unless a 
standard test oscillator is available for each type 
of circuit. Only when the crystal unit is operated 
at its series-resonant frequency or is connected 
directly in parallel with a variable capacitor, can 
the relative activity qualities of two or more crys- 
tal units be considered constant and independent 
of the particular design of the external circuit. In 
all other cases, Co, as well as R, becomes a signifi- 
cant parameter of the activity, and the exact rela- 
tion of C« to the activity will depend upon the 
circuit design. For a parallel-mode activity param- 
eter to apply in the general case, the oscillator 
circuit, itself, must be considered from a general- 
ized point of view. By this approach, the effective 

Q I^Q, =^^is often considered a more reliable 

activity quality factor, than 1/R alone. The reason 
for this belief is most readily indicated when a 
generalized crystal oscillator is represented by the 
negative-resistance method — in particular, by 
diagrams (A), (B), and (C) in figure 1-108. 

Q, AS AN INDEX OF ACTIVITY QUALITY 

1-232. If the oscillations of a crystal are to be sus- 
tained, energy must be supplied at a rate equal to 
the power losses in the crystal. This state is indi- 
cated in figure 1-108 (A), where the power source 
is represented as a generator with an emf equal. 


Section ^1 

Electrical Parameters of Crystal Units 

but opposite in sign, to the voltage across the effec- 
tive resistance of the crystal unit. The power input 
thus is |EIo| =:U*Re. Furthermore, since the total 
voltage drop around the circuit must be zero, the 
external circuit must appear as having a reac- 
tance equal, but opposite in sign, to the effective 
reactance of the crystal. Now, since the current 
through each component of the equivalent series 
circuit is the same, the voltage may be represented 
as being the result of a current flowing through 
a circuit of zero total impedance, as shown in fig- 
ure 1-108 (B) . Note that the generator is replaced 
by a negative resistance, p„, numerically equal to 
R,. Figure 1-108 (C) shows the same operating 
conditions, but with X, and of the external circuit 
replaced by an equivalent capacitive reactance, 
X„ in parallel with a negative resistance, p, equal 
X ’ X * 

to — Z,„ where Zp = ^ At all instants 

Jt«f> 

the impedance across the terminals at 1 and 2, 
whether that of the crystal unit or of the external 
circuit, is the same for both the (B) and the (C) 
equivalent circuits. Imagine now that after equi- 
librium has been reached, R,. suddenly decreases 
by one half. At this instant the power being sup- 
plied, UV. = Up, is greater than that being dissi- 
pated in R,. In (B), the amplitude of oscillations 
will increase until a new equilibrium is reached, 
at which time p. will also have decreased by one 
half and will be once again numerically equal to 
R,. In (C), the halving of R^ means that Z,, is 
approximately doubled. The same increase in cur- 
rent through Re must be shown to occur in (C) 
as in (B). Thus, the amplitude of the oscillations 
must increase until p has doubled its value and is 
again numerically equal to Z,,. (The changes in p 
are caused by the limiting elements in the oscilla- 
tor. For example, R,, of the vacuum tube will in- 
crease with an increase in gridleak bias, and this 
w'ill be reflected as an increase in p.) From the 
generalized circuit approach, we can reach the 
general conclusion intuitively that oscillations 
build up as long as |p,i > R„ or |p < Z,,, and that 
oscillations diminish in amplitude under the re- 
verse conditions. It may hot be at once apparent 
why oscillations should not build up if p were nu- 
merically larger than Z,„ in the same way that 
they do when p, is greater numerically than R,. 
A rigorous proof can be obtained by a differential 
equation of the current through the inductance, 
applying Kirchoff’s laws and keeping in mind that 
resistance, negative or positive, is mathematically 
an instantaneous rate of change of voltage with 
current. Qualitatively it can be seen that the amp- 
litude increases or decreases, depending upon 


WAOC TR 56-156 


93 



Section I 

Eiectrieai Parameters of Crystal Units 



Figure J-IOS. Generalized crystal oscillator circuit 


whether the ratio of the power input to the power 
dissipation is, respectively, greater or less than 1. 
For circuit (B) the current, Ic, is the same for 
both the crystal unit and pt, so that the power 
ratio is : 


1/ p. 


Ps 

I.*R. 


Re 


1—232 ( 1 ) 


Oscillations thus build up as long as p, is greater 
than R,. For circuit (C) the voltage, E„ is the 
same across the crystal unit as across p, so that the 
power ratio is: 


EqVp ^ Zp 
R. EoVZ.* p 


where 2^. = VRe* + X,* 
and Xe > > R, 

1—232 ( 2 ) 


Oscillations thus build up as long as p is less than 
Zp, the equivalent parallel-resonance impedance at 


equilibrium. The initial values of p, and p for a 
given frequency can be assumed to be fixed param- 
eters characteristic of the particular oscillator 
circuit, although the exact magnitudes may be ex- 
tremely complicated functions involving all the 
circuit variables. Nevertheless, it is reasonable to 
assume that the more the negative resistance must 
change, the greater will be the activity of the os- 
cillator by the time equilibrium is reached. From 
the point of view of circuit (B), it would seem 
that with a given initial p, maximum activity is to 
be obtained with a minimum Rei but in circuit 
(C), on starting with a given value of p, maximum 
activity is to be obtained with a maximum Z„. Note 
that these two conditions are not entirely equiva- 
lent. For a given crystal unit, Zp, for instance, can 
be increased by increasing X„ which in turn re- 
quires that Re, as well as X,, become greater. 
(Since X, must increase to match the increase in 
Xi, so must the frequency, and hence also R*.) 
Remembering that the activity that is assumed to 


WADC TR 56-156 


94 







be proportional to the change in negative resist- 
ance is that in the oscillator output and is not 
necessarily the current amplitude in the crystal 
circuit, it can be seen that R» alone, in spite of the 
implications to be drawn from figure 1-108 (B), 
may not be a sufficient parameter to indicate the 
relative activity quality of a crystal unit in the 
general case, i.e., X, must also be considered. For 
these reasons, the effective Q of the crystal unit, 
Qe = Xe/R„ is usually considered the more reliable 
index of the crystal activity quality for parallel- 
resonant oscillators. There are exceptions, how- 
ever, where Re, or rather is the proper activity 

parameter. These occur when the crystal is actu- 
all.v operated at series resonance with an external 
capacitance. An example is the Cl-meter circuit 
in figure 1-106, when Ci is connected in series with 
the crystal. (See paragraph 1-585 for a more de- 
tailed analysis of negative-resistance limiting.) 

1-233. The crystal Q, is a more direct index of the 
potential activity in some oscillator circuits than 
in others. The first consideration is the effect that 
a change in Q, has upon the excitation voltage. 
Normally, an increase in Qe means an increase in 
excitation, but this is not true in every case, even 
in the conventional parallel-resonant circuits. In 
these oscillators, the feed-back network may con- 
sist of a crystal unit shunted by one capacitance 
and in series with another. Referring to figure 
1-109, assume at first that the capacitance, C,, 
shunting the crystal unit in both (A) and (B) is 
negligible. The generalized circuits are thus 
equivalent to that of figure 1-108 (B). X, of figure 
1-108 (B) is represented by (Xi, -f- Xii>) and by 
(X,c X,d) in circuits (A) and (B), respectively, 
of figure 1-109. Referring now to figure 1-109 only, 
the crystal unit in circuit (A) is connected be- 
tween the control grid and cathode, so that the 
principal activity consideration is to obtain the 
desired excitation voltage across the crystal unit 
with a minimum power dissipation in the crystal 
unit. Similarly, in circuit (B) , the higher the crys- 
tal quality, the less the crystal power that would 
be required to obtain a desired excitation voltage 
across Xic. As a first approximation, assume that 
X, is much greater than R„ so that the voltage 
across the crystal unit can be assumed to equal 
IgXe, where I, is the feed-back current and where 
1,X„ is 180® out of phase with the voltage across 
the series reactance, X,., or Xj,, as the case may 
be. In circuit (A), the excitation voltage is thus 
equal to I,Xc, and the crystal power dissipation is 
1,’Re. If the ratio of the r-f output voltage of the 
tube to the excitation voltage, Ep/E„ is assumed 


Section I 

Eloctricol Paramoterf of Crystal Units 

to be k, then the ratio of the output voltage to 
the crystal power. Pc, is : 


Ep kljXe kQe Circuit (A) 

pT “ I/R, “ Ig (figure 1-109) 


The magnitude of the total impedance of the feed- 
back circuit must be kZ,, where Zg is the grid-to- 
cathode impedance. As long as it can be assumed 
that the impedance of the crystal unit is 180° out 
of phase with the reactance of the series capaci- 
tance, then the magnitude of the plate-to-grid im- 
pedance must be approximately (k -f l)Zg. Thus, 


X 

in circuit (B), |Xk.| 

value of Xu. in place of X. in the equation above: 


Ep kQ, Circuit (B) 

Pp (k -H 1) Ig (figure 1-109) 




Figure I- 109. Generalized crystal oscillator circuits, 
showing two conventional methods for connecting 
crystal unit (X^, RJ in feed-back circuit 


WADC TR 56-156 


95 



Section I 

Electrical Parameters of Crystal Units 

The equations for circuits (A) and (B) show that 
for a given k and a given crystal current, a maxi- 
mum ratio of E„/Pc is to be obtained when Q* is 
a maximum. This assumes that the circuit capaci- 
tance, Cv, directly shunting the crystal unit is neg- 
ligible. When this assumption cannot be made, the 
effective Q of the parallel combination must be 
substituted for Q^. The Q of the combination is 

approximately equal to Q,, as long as 

(X, 4- Xe) is numerically large compared with 
Re, Xy being the negative reactance of Cy. The 
larger the magnitude of the ratio Xy/X^, the more 
directly does Qe become the principal activity in- 
dex. It should be remembered that the direct pro- 
portionality between Qe and the activity in the 
example above holds only upon the assumption 
that Ig is to be held constant, regardless of the 
value of Qe. Another instance in which the activity 
of an oscillator is a direct function of Q, would be 
the unconventional case of an oscillator so de- 
signed that the crystal unit is operated in series 
resonance with an external capacitance and with 
the excitation voltage equal to, or directly propor- 
tional to, the voltage across either the crystal unit 
or the series reactance. In such a circuit, use would 
be made of the resonant rise in voltage that is de- 
veloped when a component impedance is greater 
in magnitude than the total impedance. Since the 
current through the component is the same as that 
through the total impedance, the step-up voltage 
ratio is the same as the impedance ratio. At series 
resonance the total impedance of the crystal cir- 
cuit would equal Ry (assuming no other resistance 
in the circuit) , so that if Z, of the crystal unit were 
approximately equal to X,, the ratio of the voltage 
across the crystal unit to the feed-back emf would 
be lyZy/IyRy = Q... The standard crystal units 
which are intended for use at parallel resonance 
are tested for operation with definite values of 
load capacitance, C*. Thus, the recommended oper- 
ating value of X,. may be assumed to be equal to 

= |X,I. The maximum value of R* that is per- 

missible w'ith this value of Xy is also specified. 
Hence, in the design of an oscillator that must 
operate satisfactorily with any randomly selected 
crystal unit of a given type, allowance must be 
made in the circuit design to ensure that satis- 
factory activity is obtained for the minimum 

uCxRy (max))’ 

MAXIMUM EFFECTIVE Q (Q,J 

1-234. Where it is desirable to have an activity 

WADC TR 56-156 96 


parameter that is not a function of the particular 
external load capacitance, the maximum effective 
Q (Qra.) offers a convenient index of the maxi- 
mum potential activity of a crystal unit which is 
to be operated in a type of circuit whose activity 
depends primarily upon Qe. The maximum effec- 
tive Q can be expressed solely in terms of the basic 
crystal parameters, since the maximum occurs 

midway between fr and fa, so that af Thus, 

from equations (1) and (2) of figure 1-98, it can 
be seen that 


^ ^ ^ gxLAfa 
Re 2R 

Since 


Afa 

Then 

Qem 


— [by equation 1 — 208 (1)] 

2(_/o 

2xfLC yiZ _ 1 Xco 

4RCo 4RC„ 4RaiC„ 4R 

1—234 (1) 


Qee, provides a convenient activity factor com- 
bining all the crystal parameters. It is equal to the 
maximum step-up voltage ratio that can be ob- 
tained by operating the crystal in series with a 
negative reactance. Where another capacitor, C,, 
is shunted directly across the crystal unit, the 
maximum effective Q of the combination becomes 


VLC/4R(C„ -h Cv). 


FIGURE OF MERIT, M 

1-235. In paragraph 1-216, it was shown that the 
minimum fp obtainable with a crystal unit occurs 

X 

when the apparent Q. of the motional arm, is 

equal to 2 ; that is, unless Rx, the effective resist- 
ance in the shunt arm, is significant. Within the 
frequency range at which the crystal unit appears 
as a positive reactance, the maximum value of 
Q. occurs at the antiresonant upper limit. This 
maximum theoretical value of Q, has been selected 
as a convenient figure of merit to indicate the rela- 
tive activity quality of a crystal unit, and has been 
assigned the symbol M. In general, the larger the 
value of M, the less will be the feed-back energy 
required to sustain a given activity. If M is less 
than 2, the crystal unit cannot exhibit a positive 
reactance, and hence cannot be used in conven- 
tional oscillator circuits. To sustain oscillations at 



a desired level, an oscillator will require that the 
crystal exhibit some minimum value of Q„ equal 
to 2 or greater, depending upon the oscillator, so 
that a knowledge of the M of a crystal unit is of 
Value in determining whether or not the crystal 
can be used. Formulas for M are : 


M = 



X„ 4xLAf. 2xfLC 
R R RC„ 


Q ^ y/ns 

r RC„ 


4 Qem 


1—236 (1) 


where Q is the series-arm Q and r = C„/C. Note 
that M is equal to four times the maximum Q, of 
the crystal unit, so that the measurement of either 
will indicate approximately the same performance 
characteristics. Actually, as M approaches 2 the 
value of Q,™ as given by equation 1-234 (1) be- 
comes unreliable, because of the approximations 
made in its derivation. If M = 2, Q™ is zero, 
although its approximate formula would indicate 
a value of 0.5. In practice, however, crystal units 
with such low values of M are normally far below 
specified standards, except possibly in the case of 
v-h-f crystal units operating on harmonics higher 
than the fifth, so that Q,m, which can be meas- 
sured directly as the maximum step-up voltage 
ratio obtainable with the crystal unit in series 
with a capacitor, provides a reasonably accurate 
M 

indication of-^. M was originally chosen as a fig- 
ure of merit l^cause it can be shown to be a con- 
stant of proportionality in the equation for Q„ 
and because it is expressible in terms of the crys- 
tal parameters alone. As performance parameters 
of a crystal unit, M and are practically equiv- 
alent, but M is the parameter more commonly en- 
countered in treatises discussing crystal activity. 

PERFORMANCE INDEX 
1-236. The fact that the Q, of a crystal unit is the 
most direct factor influencing the activity first be- 
came apparent through consideration of the re- 
quirements necessary for oscillations to build up 
in the generalized oscillator circuit in figure 1-108 
(C). When X, and R, are assumed to be the re- 
actance and resistance of an actual coil, it can be 
rigorously shown that oscillations build up as long 

as (CxpI < Here, Cx and p are both functions 

of the external circuit, and L, and R, can be as- 
sumed to be constants of the coil. As the amplitude 
of oscillations increases, the plate resistance of the 
tube increases, which in turn causes p to increase. 
( Cx may also vary, but usually to a much smaller 


Section I 

Electrical Parameters of Crystal Units 

degree.) Multiplying both sides by <«: |uC,p| 

or < Q«. From this point of view it would 

appear that the change in p/Xi which must be 
undergone before equilibrium is reached, or, equiv- 
alently, the rise in amplitude necessary to bring 
the plate resistance to the equilibrium point, will 
always increase or decrease with Qe, and that with 
a given R„ the amplitude increases or decreases 
with Xp. These implications can be misleading. 
First, with a given Xj, X, is no longer a significant 
variable if Q,. is equal to 10 or more, but must 
remain equal in magnitude to X,. In this event 
the only variable of the activity is Rp. Secondly, 
the rise in amplitude is more accurately a function 
of the difference between Qp and the starting value 

of rather than of the magnitude of Qp alone. 

Furthermore, Qp is, itself, a function of X,, for 
as Cx is varied from a relatively large value of 
capacitance and made to approach zero, Xp must 
increase with X,. However, Qp does not increase 
indefinitely with Xp, but reaches its maximum 
value, Qp,„, when Ci — C„ and then steadily de- 
creases as Cx becomes smaller. Yet the difference 

between Qp and the starting value of does 

continue to increase even though Q„ has passed 

X 

its maximum, for the change in is less than the 

change in the value of |pwC,|. Now, C, and even 
more so, C, (= Co -f C,), can be considered rea- 
sonably constant parameters as compared with p 
during the build-up of oscillations. Referring to 
figure 1-108 (F), it can be seen that oscillations 

build up as long as ^ > jp(for the same reasons 

X 

that hold in the case of and p/X^). Since Xt 

Xvp 

can be assumed to be relatively constant, both sides 
of the function can be multiplied by Xt, so that 

X X I 

it can be said that as long as | ^ = Zp > |/»|, 

the amplitude of the oscillations continues to in- 
crease. Thus, for a given starting value of p and 
with Xt relatively independent of the amplitude, 
the most direct index of the activity is the equiva- 
lent parallel-resonance impedance, Z,„ of the gen- 
eralized oscillator circuit. For this reason, Zp is 
called the Performance Index and has been given 
the symbol PI. PI, unlike M and Qp,„, is not a 
parameter of the crystal unit alone, but of the 
crystal unit effectively in resonance with' some 
specified load capacitance, C,. Military S andard 
crystal units intended to be operated at parallel 
resonance have a recommended load capacitance 


WAOC TR 56-156 


97 



Section I 

Electrical Parameters of Crystal Units 

specified. PI meters have been developed for meas- 
uring the performance index directly, but there 
are very few such meters available. Where the PI 
of a crystal unit is desired, it can readily be com- 
puted from measurements made with standard Cl 
meters. Various expressions of PI are given below : 

PI = ^ = 1 ^ LC 

R a," (Co -f- C.)" R RCt® 

I Xco ^ I MXco I 

Note that PI is not a function of the crystal alone, 
but of C, as well, and care should be taken that 
the capacitance ratio of C,/C„ is not mistaken for 
r = Co/C. It can be seen that the maximum PI 
occurs at antiresonance, where Ci = 0, so that 
(max) PI = |MXco|. 

1-237. The PI of a crystal unit, or more properly, 
of a crystal circuit is usually found to be an im- 
portant parameter entering the equations of an 
oscillator circuit, particularly when the equations 
express the conditions required for a given out- 
put. As a simple example, consider again the two 
generalized oscillator circuits in figure 1-109. In 
paragraph 1-233, it is shown that the ratio of 
to crystal power, P^, is equal to kQe/Ig for circuit 
kQ 

(A), and equal to ‘ , for circuit (B). As 

long as Ij is considered predetermined, the volt- 
age-to-power ratio is primarily a function of k 
and Q,.. Normally, however, it is not Ig that is to 
be predetermined, but rather the crystal power 
that must not exceed the maximum value. When 
Z, X,., the magnitude of the impedance of the 
feed-back arm in circuit (A) can be assumed to 


be equal to kX^, and to 


kX, 


k 1 

is thus equal to Ep/kXe in (A) and to 
in (B). On substitution in the Ep/P^ equations: 

k^QeXe 


in circuit (B). Ig 

E„(k + 1) 
kX. 


or 


Ep/Po 


EpVPc = 


Ep 

R» 


, Ep^ k^PI 

Similarly, — = 


= k^PI Circuit (A) 
Circuit (B) 


Inasmuch as the power output of the oscillator can 
be assumed to be directly proportional to Ep*. then, 
for a given drive level of the crystal unit, the 
power output will vary directly with the PI. It 


should be noted that the EpVPc equations above 
are not affected by the assumption of shunt ca- 
pacitances, Cy, across the crystal unit. Just as 

X * X * 

PI = " 5 ^ — the same value holds even if the 

XVe 

shunt capacitance is assumed to be increased by 

Cy. 

Frequency Stabilization Quality of Crystal Unit 

1-238. The over-all frequency stability of a crystal 
oscillator is dependent upon the stability. of all the 
parameters influencing the crystal circuit; these 
in turn are dependent upon the stability of the 
power source and the load, as well as the ambient 
conditions under which the oscillator is required 
to operate. The over-all stabilizing ability of the 
crystal is dependent upon both the stability of the 
crystal parameters when the crystal is exposed to 
changes in temperature or drive level, and the 
ability of the crystal to minimize the change in 
frequency that is necessary when the parameters 
of the external circuit deviate. It is this latter 
quality of the crystal that makes the crystal os- 
cillator superior to oscillators that use only coils 
and condensers to control the frequency, and is the 
type of frequency stability that concerns us now. 
1-239. The frequency stabilizing property of a 
crystal is normally expressed as the rate at which 
its reactance changes with frequency. In figure 

dX 

1-110 are shown the curves of -rj for the series- 

dt 

arm parameters L and C. Resonance happens to 
occur at the frequency at which Xc is changing at 
the same rate as Xf Since the rate of change of 
Xi, is a constant, at frequencies i jar resonance 
it can be said that the total change in reactance 
with frequency is primarily a function of L, 
for the absolute rate of this change is the same 
whether C is large or small. Normally, however, 
it is not the absolute change in reactance that is 
important — it is, rather, the change in reactance 
per percentage change in frequency, or, more usu- 
ally, the percentage change in reactance per per- 
centage change in frequency. When the frequency 
stability is expressed in percentage, it is no longer 
primarily a function of L, but becomes dependent 
upon the other crystal parameters as well. Only 
where the major concern is to produce a definite 
shift in reactance or frequency for a given change 
in the external circuit does the major attention 
center upon the parameter L. Just as the relative 
activity potential of a crystal depends somewhat 
upon the type of circuit in which it is used, so 
also does the frequency stabilizing characteristic 
of a crystal depend upon the external-circuit de- 


WADC TR 56-156 


98 




f, TO U 


Figure I* 1 10. Rtites of change of reactance of equiv- 
a/ent series-^rm parameters, I and C, with frequency 

sign. A relative stability index will be discussed 
briefly for each of three general types of circuits ; 
where the crystal is operated at its normal series- 
resonant frequency, where it is operated in paral- 
lel with a negative reactance, and where it is 
operated in series with a negative reactance. 

FREQUENCY STABILITY AT 
SERIES RESONANCE 

1-240. Since the total series-arm reactance at f, 
is equal to zero, it is not convenient to express the 
relative frequency stability in terms of the per- 
centage rate of change in reactance. Approxi- 
mately the same considerations apply for the reso- 
nance frequency, f,. Also, the effective stability in 
a given circuit may well depend more upon the 
rate of total impedance change or the rate of phase 
shift with frequency than upon the actual rate at 
which the reactance changes. Suppose, for exam- 
ple, that the feed-back energy must pass through 
the crystal unit and return to the oscillator input 
in a certain phase. If, because of a change in the 
circuit parameters, the feed-back energy is re- 
turned slightly out of phase, the frequency will 
have to shift away from the normal resonant point 
exactly enough for the crystal to correct the 
change in phase. If the change in phase has orig- 
inally been caused by a change in the reactance of, 
say, a capacitor connected directly in series with 
the crystal, it is only necessary for the frequency 
to shift the amount necessary for the crystal re- 
actance to exactly counteract the change in the 
series reactance. In this case, the resistance of the 
crystal circuit is not effective in degrading the sta- 
bility. It is true that the greater the resistance 
that the crystal faces, the greater must be the fre- 


Section I 

Electrical Parameters of Crystal Units 

quency change to produce a given phase shift. But 
on the other hand, since the series capacitance 
faces the same resistance, the initial phase shift 
due to a change in the capacitance is correspond- 
ingly reduced. In this case, the frequency stability 
of one crystal as compared with that of another 
depends almost entirely upon its relative rate of 
change of reactance. At series resonance the fre- 
quency stability factor can be defined as 

j Y 

F, = — = 2coL = 2 VlTC 1-240 (1) 
df 

where F, is the rate of change of reactance per 
fractional change in frequency. Thus, in compar- 
ing one crystal unit with another, the one with the 
larger L/C ratio can be assumed to provide the 
greater frequency stability at series resonance. 
However, if the change in reactance occurs at a 
point in the circuit only loosely coupled to the 
crystal, the resistance of the feed-back circuit is 
relatively ineffective in reducing the phase shift of 
the feed-back voltage, but instead, tends to in- 
crease the change in frequency necessary for the 
crystal to correct the phase. In the case of a feed- 
back network where the crystal must compensate 
for a change in phase that is relatively independ- 
ent of the resistance in the crystal circuit, the fre- 
quency stability is more directly measured by the 
rate of change of phase in the crystal circuit as a 
whole than by the rate of change of reactance 
alone. 

1-241. Figure 1-111 shows that a small phase 
displacement, A(9, at series resonance is approxi- 
mately equal to aX,,/R.., where aX, is a small 

© Zc ' Sc +iO(WHEN ( = fr ) 

@ Zo = Rc +jAX,(WHEN f = fr +Afl 



Figure I-III. Phaser diagram, showing change in 
reactance, AX,, of series-mode crystal required to 
produce a small change in phase, AS, where the re- 
sistance of the crystal circuit is equal to R,, 


WAOC TR 56-156 


99 



Section I 

Electrical Parameters of Crystal Units 

change in the effective reactance of the crystal, 
and R,. is the total effective resistance in the crystal 
circuit (equal to R^ of the crystal plus R, of the 
external circuit). For convenience, the frequency 
is usually expressed in terms of angular frequency, 
lu = 27ff radians per second, instead of cycles per 
second. In equations (1) and (2) of figure 1-98, 
it can be seen that for small values of Af, the de- 
nominators of the approximate equations of X, 
and Re are approximately equal to unity, in which 
case Xe !=» 2 LAco and R, «=■ R. Thus, 


^ „ AXe 2LAca> 

^ ~X~ " R + R. 

The frequency-stability index can be defined to be 


dw R -j” Rj 


1—241 (1) 


Expressed as the change in phase angle per percentage 
change in frequency, equation (1) becomes: 

axid __ 2<»)L 

100 dw "" 100 (R + R.) 


or more simply: 


wdS 

du 


2coL 

X 


2X, 2 ^ 


1—241 (2) 


The last term on the right shows that where the 
fractional rate of change is concerned, the fre- 
quency stability is directly proportional, not sim- 
ply to L, but to the square root of the L/C ratio. 
Equation (2) also shows that the frequency 
stability is inversely proportional to the crystal 
circuit resistance. But it must be remembered that 
this is true only to the extent that the original 
phase shift of the input to the crystal circuit can 
be considered independent of Re- 
1-242. As an exaggerated example, we can see 
that minimum stability is to be expected if the 
input to a high-resistance, series-resonant, feed- 
back circuit is supplied through a weak coupling 
from a plate tank circuit sharply tuned to the reso- 
nant frequency but having an impedance small 
compared with the Rp of the tube. Since a slight 
change in the parameters of the tank circuit could 
shift the phase of the feed-back input almost 90 
degrees, such an oscillator would obviously be 
completely unstable, even if it were assumed that 
oscillations could be maintained. 


FREQUENCY STABILITY AT 
PARALLEL RESONANCE 

1-243. When it can be assumed that a crystal unit 

WADC TR 56-156 100 


is effectively operating in parallel with an external 
capacitance, C„ the value of which is relatively 
independent of small changes in the frequency, 
the frequency stability can be assumed to be di- 
rectly proportional to the rate of change in the 
reactance of the motional arm of the crystal. In 
this case, it would seem that a frequency-stability 
factor for parallel resonance 

Fp = ^ ■ 0, = 2 ViX = F, 

do) 

would be appropriate — ^just as in the case of series 
resonance — and Fp would be identical with F,. 
However, it will be recalled (see figure 1-103) that 
the higher the reactance, i.e., the smaller the value 
of Ct, the less stable will the oscillator become. 
Taking this into account, a more accurate indi- 
cation of the stabilizing quality of a parallel-reso- 
nant crystal is given by what is called the fre- 
quency-stability coefficient, which is the percent- 
age rate of change in reactance for a percentage 
change in frequency. Thus, 

dX. 100 _ 2a)L f 

^ " d« ■ X. ■ 100 “ X. Af 

This result is quite interesting, for it indicates that 
for crystal units of the same frequency equal sta- 
bilities can be achieved simply by operating the 
crystal units at the same value of Af above series 
resonance. Since the same equation holds for any 
value of L and C, it is not immediately apparent 
as to why a crystal is so much more stable than 
a conventional inductor and capacitor. In para- 
graph 1-208 it was found that the ratio is 

Q 

equal to If Afp is substituted for Af„ and Ct 
is substituted for C,, then 


Fx. = 


f 2 Ct 
Afp " C 


It can be seen that in order for a conventional 
series-parallel inductor-capacitor network to have 
the same theoretical stability as a crystal, the 
shunt capacitance must be thousands of times 
greater than the series-arm capacitance. The 
series arm of such a network would require an 
extremely small L/C ratio. The parallel impedance 
would be small, and the net series-arm reactance 
much smaller still. The crystal, on the other hand, 
has such a small value of C that the reactances are 
reasonably large even for small values of Af. Al- 
though the equation for the frequency-stability 
coefficient indicates an unlimited stability if Afp 
is simply made small enough, this would be theo- 



retically true only for a circuit resistance equal 
to zero. In pratice, as X, approaches the motional- 
arm R in magnitude, a given change in Xt will 
cause a greater change in the phase of the over-all 
circuit impedance than will the same change in 
X.. Of significance is the fact that the frequency- 
stability coefficient, Fx., represents the stabilizing 
effect of a crystal for the percentage change in the 
reactance of the total effective shunt capacitance, 
Ct (= Co -f- Cx). With C„ considered constant, a 
given percentage change in the total capacitance 
becomes less than the actual percentage change 
in the variable component, which we can assume 
to be the equivalent external capacitance, C,. For 
a given Ct, the larger the ratio C„/Ci the smaller 
will be the percentage change in Ct for a given 
percentage change in Cx, and the greater will be 
the oscillator stability in the face of changes in 
the external circuit. In other words, the effective 
frequency-stability coefficient of the crystal unit 
as a whole is greater than that of the motional- 
arm alone if it can be assumed that the percentage 
changes in C„ will be negligible compared to those 
in C,. When R is small compared with X„ X^ «== 
XcoX,/(Xro + X,) and the effective frequency- 
stability coefficient becomes: 

^ ^ 2LXco^ a»(Xco -f X.) 

do, ■ X. (Xco + ■ Xco X. 

2ci>Ij Xco fCx 

^ X ■ Xc„ -t- X. ^ AfX 

Since 


then 

Ct 2Ct* 

Fx. = Fx. • ^ 1-243 (1) 

It should be remembered that equation (1) is 
based upon the assumption that C, is constant 
and that any change in X, will be due to a change 
in C,. If Co is effectively increased by a fixed ca- 
pacitance, Cv, directly shunting the crystal unit, 
the effective variable C, becomes smaller. The 
effective C„ insofar as the frequency stability is 
concerned, will equal Ct — (C„ -f- C, ) . Substituting 
this effective value of C. in equation (1) will pro- 
vide a more accurate frequency-stability coefficient 
for a crystal unit directly shunted by a fixed C,. 


Section I 

Electrical Parameters of Crystal Units 

FREQUENCY STABILITY AT SERIES RESO- 
NANCE WITH EXTERNAL CAPACITANCE 

1-244. The effective frequency-stability coefficient, 

20r* 

Fx,. = provides an appropriate index of the 

frequency-stability quality of a crystal unit oper- 
ated in series resonance with an external capaci- 
tance Cx, for the same reasons that make the co- 
efficient applicable in the case of parallel-resonant 
circuits. Ct, here, represents the sum of two actual 
capacitances, Co -f- C„ and has a more concrete 
meaning than simply a generalized parameter. Fx. 
gives the percentage change in X,. per percentage 
change in frequency. The reciprocal, 1/Fx„, can be 
interpreted as equaling the percentage change in 
frequency that will occur per percentage change 
in the negative reactance Xj. In unconventional 
circuit designs, where a significant phase shift 
can occur as a result of changes in the impedances 
in the oscillator output circuit {which is only 
weakly coupled to the feed-back input), the resist- 
ance of the feed-back circuit may need to be taken 
into account in a manner similar to that discussed 
in the case of crystal units operating in series 
resonance. 

FREQUENCY STABILITY OF 
OVER-ALL CIRCUIT. 

1-245. Although the absolute values of the fre- 
quency-stability indices discussed in the foregoing 
paragraphs depend upon generalized parameters 
of the external circuit, the values are primarily 
useful for indicating the relative stabilization qual- 
ity of one crystal unit as compared with another 
when operated under similar circuit conditions. 
The actual frequency drift due to changes in the 
supply voltages, tube characteristics, circuit im- 
pedances, etc., depends upon the particular oscil- 
lator design as well as upon the performance 
characteristics of the circuit components. The 
percentage variation that can be expected in the 
generalized parameter, Cx, is of equal importance 
in gauging the frequency stability of the oscillator 
as a whole. The crystal-unit frequency-stability 
coefficients appear as single parameters among 
others in the frequency-stability equations for each 
particular type of oscillator circuit. In general, the 
series-resonant type of oscillator has the greater 
frequency stability, permitting tolerances from 
four to twenty times as narrow as those normal 
for parallel-resonant oscillators. Indeed, in a well- 
designed series-resonant oscillator where the re- 
active components are negligible in their effect on 
the phase of the feed-back voltage, the frequency 
stabilization of the crystal unit can be very nearly 


WADC TR 56-156 


101 



S«ction I 

Electrical Parameters of Crystal Units 


perfect, so that the most significant factor to con- 
sider is the stability of the equivalent-circuit 
parameters of the crystal unit, itself. One source 
of frequency instability common to both series- 
resonant and parallel-resonant oscillator circuits is 
the presence of harmonics in the output. For cer- 
tain applications, such as in crystal calibrators, 
these harmonics are desirable, but in most cases 
it is preferable that they be kept to a minimum. 
Harmonics are unwanted not only for the sake of 
a sine-wave output as such, but also because they 
introduce reactive components in the crystal cir- 
cuit, thereby increasing the chances of frequency 
instability. The harmonics can be reduced by de- 
signing the oscillator plate circuit to provide a 
low-impedance bypass path for them, and by using 
low plate and grid voltages. Unwanted reactive 
effects in the oscillator circuit also occur as a result 
of feedback from the amplifier stages following 
the oscillator. These can be minimized by the use 
of proper shielding, buffer amplifiers, neutralizing 
circuits, and by careful attention to the physical 
layout in designing the equipment, to ensure that 
all leads are as short as practicable and that 
the oscillator is electrically isolated from circuits 
carrying high amplitudes of r-f voltage or current. 
The effective load capacitance, Cj, with which the 
crystal unit resonates is usually a function of the 
vacuum-tube parameters, the load resistance, the 
effective grid resistance, as well as the reactive 
impedances in the feed-back and output circuits. 
All these variables are, in turn, functions of the 
oscillator output and the grid and plate d-c volt- 
ages. Thus, the frequency stability is dependent 
upon the degree of voltage regulation, the con- 
stancy with which the oscillator load is main- 
tained, and in the care taken in the original design 
to ensure that the circuit components are so pro- 
portioned that the effects of variations in the tube 
parameters are minimized. Silvered mica capaci- 
tors normally are to be preferred for fixed capaci- 
tances in the tuned circuits. Those capacitors hav- 
ing dielectrics composed of titanium compounds 
can be used for r-f bypass purposes, but are too 
variable under changes in temperature and voltage 
for use as tuning components. Air-dielectric ca- 
pacitors are almost always adjustable. With the 
exception of the vacuum-dielectric capacitor, the 
air-dielectric type is the most stable and is to be 
preferred for small capacitances and variable tun- 
ing elements. As a rule, the improvements in 
circuit design that permit of greater frequency 
stability necessitate additional circuit components, 
additional tuning adjustments, narrower operating 
frequency ranges, smaller voltage or power out- 
puts, or some combination of the above. 


Bandwidth and Selectivity Parameters 
of Crystal Unit 

TOE CAPACITANCE RATIO, r 

1-246. The bandwidth of a crystal unit refers to 
the particular frequency range over which the 
crystal unit can be operated in a given oscillator, 
filter, or transducer circuit. In the case of a con- 
ventional oscillator circuit, the applicable fre- 
quency range is that in which the crystal can 
appear as an inductive impedance. In cycles-per- 
second, this range is Af, = f. — f,. Percentage- 
wise, the bandwidth is , which, as shown 

Ip 

in paragraph 1-208, is approximately equal to 

^ Although the practicable operating 

range does not extend over the entire band, it can 
be seen that the relative merit of a crystal unit 
insofar as its range of frequency adjustment is 
concerned can be indicated inversely by the param- 
eter r, whereas the relative selectivity is indicated 
directly by r. In figure 1-95, it can be seen that 
the smallest theoretical values of r (when the dis- 
tributed capacitance is negligible, so that r = r,) 
are obtained with the low-frequency, length- 
extensional-mode elements of the X group. The 
smallest capacitance ratios are provided by ele- 
ment E, which has values of r as low as 120 to 
125. These are equivalent to a resonance-to-anti- 
resonance bandwidth on the order of 0.4 per cent 
of the nominal frequency. For the high-frequency 
A and B elements, the bandwidths are approxi- 
mately 0.2 and 0.083 per cent, respectively. 

1-247. Insofar as frequency control is concerned, the 
resonance-to-antiresonance bandwidth is impor- 
tant primarily as a relative index of the frequency 
range through which a parallel-resonant oscillator 
can be made to operate by varying the load capaci- 
tance, Cx. For example, the tuning adjustments of 
an oscillator employing an A element can vary the 
frequency approximately two-and-a-half times as 
much as can the same adjustments if the oscillator 
employs a B element. Although small frequency 
adjustments are possible, the high selectivity of 
quartz crystals precludes their use in frequency- 
modulated oscillators. Eventually, it may be that 
crystal units mounting high-frequency EDT plates, 
which have capacitance ratios as low as 20, will 
find an application in this field, but at the present 
time EDT crystals are used almost exclusively in 
filter networks. As a filter element, the capacitance 
ratio of a crystal is of greater importance than in 
frequency-control circuits. Filter networks, com- 
posed of crystal units alone, can be designed for a 


WADC TR 56-156 


102 



Section I 

Electrical Parameters of Crystal Units 


maximum pass band of - ■ , which in the case 

Ip 

of quartz means a maximum pass band of 0.8 per 
cent. For the low-frequency networks, such as are 
normal to telephone carrier systems, this is much 
too selective for passing voice channels. For this 
Peason, quartz crystals employed in 1-f telephone 
carrier filters must be used in conjunction with 
inductors and capacitors. The narrow bandwidths 
of quartz elements used alone are primarily appli- 
cable in filters when it is desired to pass a single 
frequency, such as the pilot signal of a carrier 
system. 


ELECTROMECHANICAL COUPLING 
FACTOR, k 

1-248. To the extent that the equivalent circuit of 
figure 1-91 is applicable it can be assumed that 
when a crystal unit is connected across the ter- 
minals of a battery the ratio of the energy stored 
in electrical form to the energy stored in mechan- 
ical form is equal to the capacitance ratio r = 
Co/C. The electrical energy is that stored in the 
static capacitance, Co, and is equal to V’Co, 
where V is the applied d-c voltage. The mechanical 
energy is the energy that is stored because of the 
piezoelectric strain in the crystal, and is equal to 
1/2 V*C. In transducer applications, it is useful to 
rate a crystal according to the ratio of stored 
mechanical energy to total applied electrical energy 
under the conditions of d-c or very-low-frequency 
applied voltages. The parameter for this purpose 
is the electromechanical coupling factor, k, equal 
to the square root of the ratio of the stored 
mechanical to the total input energy. As such, k is 
an index of the crystal efficiency as a transducer. 
This factor is given by the formula 


k = 



1—248 (1) 


where c is the dielectric constant, s is the elastic 
compliance, and d is the piezoelectric constant giv- 
ing the ratio of strain to field. According to the 

energy ratio, k* should equal 7 ; — or approxi- 

O© “P V 

1 c 

mately -. However, the capacitance ratio, at 

resonance can be shown to be ^ times as large as 

the theoretical ratio at zero frequency. Actually, 
then the ratio is 


stored mechanical energy 
total stored electrical energy 


!L“ 9 . 
8C. 

1—248 (2) 


where C and Co are the equivalent capacitances at 
resonance. Since the bandwidth is proportional to 
C/C„, so also is it proportional to k*. In transducer 
applications, when an inductor is shunted across 
the crystal to tune out the electrical capacitance, 
and the crystal is operated near resonance, up to 
90 per cent efficiency is possible in the conversion 
of electrical to mechanical energy. Under these 
conditions, k is not a direct index of the transducer 
efficiency, but it does serve as a parameter for 
estimating the frequency range over which the 
efficiency is 50 per cent or greater. The ratio of 
the highest to the lowest frequency for greater 
than 50 per cent conversion is: 


Crystal Quality Factor, Q 

1-249. The quality factor of a crystal unit is the 
Q of the motional arm at resonance. Thus, 



Quartz crystal units are obtained with Q’s rang- 
ing in value from 10,000 to more than 1,000,000. 
The Q is a performance parameter that provides 
an indication of the ratio of the stored mechanical 
energy of vibration to the energy dissipated in the 
crystal unit per cycle at resonance. If I, is the 
r-m-s current through the series arm at resonance, 
then, at the instant the current is a maximum, the 
equivalent capacitance C can be assumed to be 
completely discharged and all the vibrational en- 
ergy, Ev, is at that instant in kinetic form. This 
energy is equivalent to that stored in motional-arm 
• inductance, L. Therefore, 


E. = = CL 

2 


1—249 (1) 


The energy dissipated per second, P^., is I.’R. Thus, 
the ratio of the stored mechanical energy to the 
energy dissipated per second is 


E, _ I.*L _ L 
Pe ~ I."R “ R 


1—249 (2) 


It can be seen that for a given wattage, the greater 
the L/R ratio the greater will be the amplitude of 
vibration. Regardless of the wattage, for a given 
L the amplitude of vibration will vary approxi- 
mately directly with the current. Theoretically, 


WADC TR 56-156 


103 



Section I 

Electrical Parameters of Crystal Units 

since there are no Military Standards setting a 
minimum limit for the series-resonance resistance, 
a crystal unit can be so excellently mounted that 
it would be vibi-ated near its elastic limit if atten- 
tion were given only to the power dissipation 
rather than to the current. Such a situation is not 
likely to arise except possibly in the case of a 
crystal-controlled power oscillator, where space 
and cost limitations require a crystal drive level 
far in excess of the rated level. More important 
from the point of view of maintaining a sinusoidal 
wave shape of the excitation voltage and of im- 
proving the stability of the oscillator is the ratio 
of the stored energy to the energy dissipated per 
cycle, rather than per second. In terms of angular 
frequency, the dissipation per radian is I.’R/w, 
so that 


Ey _ 

P./u) " I/R 


1—249 (3) 


In an actual series-resonant circuit, it is the Q of 
the entire circuit rather than of the crystal unit 
alone that must be considered, so that R should 
be replaced by the total circuit resistance. If a 
tuned, class-C-operated circuit is to be effective in 
maintaining a sinusoidal wave shape and in re- 
ducing harmonics, the energy stored in the circuit 
should be at least twice the amount that is dissi- 
pated over the entire cycle. That is. 


(min) 


Ey 

Pc/f 


2r 


2 


This requirement is met easily in quartz-crystal 
circuits, but it is an important consideration in 
the design of plate tank circuits that are to be fed 
in pulses not smoothed by the action of the crystal. 
The crystal Q is also an important parameter in 
crystal filters. In general, the higher the Q the 
sharper the pass band. 

1-250. Since the Q of a crystal is equal to 

and since the ^ ratio for a given frequency can 

be increased to almost any value desired by de- 
creasing the electrode area and by orienting the 
crystal in a direction of weak piezoelectric effect, 
or by using twinned crystal blanks, it might be 
wondered why much larger values of Q are not 
in use. The reason is that the L/C ratio and the 
equivalent series-arm resistance of the crystal are 
not independent of each other. As \/L/C increases, 
so also does R. This can be intuitively seen if it 
is kept in mind that fundamentally the Q is the 
ratio of the energy stored to the energy dissipated 


per angular cycle. Suppose that we have two 
crystal plates, A and B, both of approximately 
the same size and normal frequency, and both 
mounted exactly alike in that the frictional losses 
of one are the same as those of the other for the 
same energy of vibration. We shall also assume 
that these mechanical losses account for most of 
the crystal driving power. In other words, we 
are assuming that the two crystals have approxi- 
mately the same quality factor, Q. Now, suppose 
that crystal A has a much smaller electrode area 
than does crystal B, or that for some other reason 
the piezoelectric effect of A is very weak com- 
pared with that of B. Under these conditions, 
crystal A will have a much larger equivalent L/C 
ratio than does crystal B. But since the Q of A 
equals the Q of B, it can be seen that the series- 
arm R of A must be greater than the series-arm 
R of B in the same proportion as the square roots 
of the respective L/C ratios. It should be under- 
stood that the Q and the L/C ratio are compara- 
tively independent variables as far as R is con- 
cerned. Where R could not be estimated without 
a knowledge of Q and L/C, the latter theoretically 
could be approximated separately and independ- 
ently by an examination of the fabrication of a 
crystal unit. L and C, for example, are approxi- 
mately predetermined by the electrode area and 
the type and size of the crystal element. The Q is 
also to a certain degree a function of the same 
variables, but for given internal frictional prop- 
erties, is primarily determined by the quality of 
the crystal finishing and mounting. Thus it is that 
the Q is largely determined by the frictional losses 
and is not subject to control by varying the L/C 
ratio. Indeed, as the L/C ratio increases, the piezo- 
electric effect can become so weak and the resist- 
ance so high that the crystal cannot be shocked 
into oscillation unless very high voltages are em- 
ployed. Once in oscillation, a high L/C crystal unit 
could presumably operate satisfactorily, except 
that only very small currents could be withstood 
without the cr.ystal shattering or arcing. There is 
a hypothetical case where an exceptionally large 
L/C ratio could be practical. Such a situation 
would arise if for any reason the external circuit 
resistance faced by a crystal could not readily be 
reduced below some large minimum value. In this 
event, the use of a crystal unit of normal Q but 
large L/C ratio would prevent the over-all circuit 
Q from being excessively degraded by the external 
resistance. Ordinarily the selection of a crystal 
unit will be made on the basis of considerations 
other than the L/C ratio, but where all else is 
equal, including the average values of Q, it might 
be assumed that the crystal units having the some- 


WADC TR 56-156 


104 



what higher values of L/C are generally more 
suitable for those oscillators which do not require 
a crystal to sustain oscillations, but only to stabi- 
lize them. Such a circuit oscillating at or near 
the crystal frequency can build up the crystal 
vibrations over a large number of cycles of small 
amplitude, thereby obviating the need of large 
voltage surges or abnormally high vacuum-tube 
amplification. 

Stability of Crystal i*arameters 

1-251. Regardless of how well designed a crystal 
oscillator may be, or how high the degree to which 
the crystal stabilizes fluctuations in the external 
circuit, the over-all performance will depend upon 
the stability of the crystal parameters, themselves. 
Changes in the crystal parameters are primarily 
due to aging, changes in the ambient temperature, 
spurious modes, and to changes in the drive level. 
Aging, here, is used in its broadest sense to include 
practically any nonreversible change in the crystal 
characteristic from whatever cause. The principal 
causes and effects of aging are discussed in para- 
graphs 1-172 through 1-181. 

EFFECT OF TEMPERATURE UPON 
CRYSTAL PARAMETERS 

1-252. The temperature-frequency characteristics 
of quartz plates are covered in the description of 
the various elements, and- will not be repeated here 
except to note that a change in the frequency 
means a change in the LC product of the motional 
arm. To what extent the frequency drift may be 
due to a change in L and to what extent to a change 
in C would require very precise measurements of 
f, and fp, and. the approximate formula in para- 
graph 1-225 for computing C from the measured 
parameters would need to be replaced by a more 
rigorous equation. Although of theoretical value, 
such small changes in L or C are not, in them- 
selves, of practical importance in circuit design — 
rather it is the change in the LC product (i.e., in 
the frequency) which is important, and which 
must be kept to a minimum. Low-temperature- 
coefiicient crystals have been developed for this 
purpose, but only the GT, at low frequencies, and 
the AT, to a lesser extent, at high frequencies 
provide a near-zero coefficient over a wide tem- 
perature range. The more exacting the require- 
ments, the more expensive the crystal unit will be. 
Fortunately, zero coefficients can be obtained at 
different temperatures by slight variations in the 
orientation angle of the cut. By mounting the crys- 
tal in an oven thermostatically controlled near the 
zero-coefficient point of the crystal, the tempera- 


Section I 

Electrical Parameters of Crystal Units 

ture effected frequency deviation can be kept very 
small. Indeed, an ideal oven having a zero tem- 
perature fluctuation would permit any type of 
quartz cut to be stable provided the drive level 
remained constant. Nevertheless, the use of an 
oven is to be avoided where possible, because of 
the additional cost, space, weight, and power re- 
quirements, and also because the crystal pins of 
the oven increase the shunt capacitance across the 
crystal. The additional shunt capacitance proves 
increasingly objectionable at the higher frequen- 
cies, and makes it necessary that either the oven 
dimensions be as small as possible or that the 
entire oscillator be mounted within the oven. 
Either requirement serves to reduce the stability 
of the oven temperature, particularly if the am- 
bient temperature varies between wide extremes. 
For ovens of practical size and construction, some 
frequency deviation is to be expected as a result 
of temperature changes. If this deviation is to be 
kept to an absolute minimum, precise tempera- 
ture-coefficient characteristics must be specified in 
selecting a crystal unit, or a greater precision in 
temperature control than is now attainable in the 
average crystal oven must be sought. An in- 
genious method of obtaining practically a zero 
temperature coefficient for A elements over a 
span of 20°C and more is being developed by the 
Hunt Corporation. During a luncheon conversation 
several years ago between E. K. Morse, S. Ryesky, 
and D. Neidig (the former, a Government repre- 
sentative, the latter two of Hunt) concerning the 
possibilities of improving the frequency stability 
of Radio Set AN/ARC-1 in its first modification, 
the idea originated of operating two temperature- 
compensating equal-frequency A elements in 
series. The angles of cutting could be so selected 
as to provide equal temperature coefficients of op- 
posing polarities which would cancel when both 
crystals were operating at the same temperature. 
However, little was attempted in this field until 
recently. Experimental models show^ that over 
room-temperature ranges the frequency deviation 
can be quite negligible. Aging data* is still insuffi- 
cient, but over a period of six months a stability 
of about one-half part per million has been 
achieved, with three-fourths of the drift occurring 
in the first three months. Probably the most sig- 
nificant recent activity in the development of 
fabrication processes designed to stabilize the 
crystal parameters against changes in tempera- 
ture centers around the current investigations 
under the direction of Dr. E. A. Gerber of the 
Signal Corps Engineering Laboratories. As re- 
ported by Mr. D. L. Hammond in a modest paper. 
Effects of Impurities on the Resonator and Lat- 


WAOC TR 56-156 


105 



Section I 

Electrical Parameters of Crystal Units 

tice Properties of Qvurtz, presented at the 1955 
Signal Corps Frequency Control Symposium, a 
systematic exploration is under way to discover 
and catalog the effects on the parameters of quartz 
crystals which have been synthetically grown to 
include controlled percentages of impurities. Im- 
purity elements being experimented with include 
aluminum, boron, calcium, germanium, lead, se- 
lenium, tin, titanium, and zirconium. This work 
undoubtedly has revolutionary possibilities. The 
discoveries already made presage the probability 
that temperature effects, which are now so im- 
portant a problem, can in the future be largely 
eliminated by growing crystals, for particular 
cuts, with controlled impurities of proper quanti- 
ties and proportions. 

1-253. A crystal operated at an overtone mode 
will have temperature-frequency characteristics 
different from those exhibited by the same crystal 
at its fundamental vibration. For the control of 
very high frequencies the A element is normally 
preferred to the B element, because of its stronger 
piezoelectric effect and because of the smaller fre- 
quency deviation possible for large variations in 
temperature. However, an AT cut ideally oriented 
for operation at the fundamental mode is not 
usually ideally oriented for the higher modes. A 
research team at Philco Corporation investigat- 
ing the characteristics of harmonic-mode crystals 
found that by far the greatest change in the tem- 
perature-frequency characteristics of A elements 
occurs at the first operable harmonic jump, i.e., 
between the fundamental and the third harmonic. 
(See figure 1-112.) Since the subsequent changes 
at the higher harmonics are relatively small, a 


crystal suitably oriented for the fifth harmonic 
will usually be suitable for operation under the 
same temperature conditions at all other over- 
tones. The sensitivity of the crystal to slight 
changes in the orientation angle is acute. Figure 
1-113, for example, shows the degree by which 
the characteristic curve of an llth-harmonic A 
element is rotated by successive changes in the 
orientation angle of only 3 minutes each. If this 
crystal were to be operated at room temperature, 
an orientation of approximately 35°27' would 
appear to be preferred. For operation under 
temperature variations of — 55° to +70°C, an 
orientation of 35°30' permits the minimum total 
frequency deviation from a room-temperature 
mean. Finally, if the crystal is to be mounted in 
an 85° crystal oven, an orientation of 35°33' 
would be optimum. 

EFFECT OF SPURIOUS MODES UPON 
CRYSTAL PARAMETERS 
1-254. Closely allied with the problem of tempera- 
ture control is the problem of avoiding spurious 
modes. Spurious modes are most apt to occur in 
the case of thickness-shear crystals. Among these 
elements, the AC and BC cuts provide the purest 
frequency spectrum, but unless crystals of these 
types are provided with precise temperature con- 
trol their larger temperature coefficients prevent 
their being preferred over A and B elements. Cut- 
ting the crystal blank to the proper face dimen- 
sions is the most important factor in avoiding 
unwanted modes, but even when due precautions 
are taken, sudden apparent variations in the mo- 
tional-arm parameters of individual crystal units 



-45 -25 -5 +5 +25 +45 +65 +85 

CRYSTAL TEMPERATURE .CENTIGRADE DEGREES 


Figure I-II2. Typical variations in frequency-temperature characteristics of A element 
when operated at different harmonics 


\ 


WADC TR 56-156 


106 


are hot uncommon. These effects occur most often 
during variations in temperature, and are due to 
the fact that the temperature coefficients of nearby 
modes are quite high. The activity and frequency 
curves Versus temperature of an erratic A element 
at series resonance in a tuned bridge circuit are 
shown in figure 1-114. The activity was measured 
by the grid current. No tuning adjustments were 
made during the temperature run. Note that the 
sudden jumps occur at some of the same frequen- 
cies, which, at the high-temperature portion of 
the curve, are apparently of a reasonably pure 
mode, indicating that the temperature coefficients 
of the desired and the unwanted modes are differ- 
ent. It can also be seen that the sudden dips in 
frequency are accompanied by abrupt changes in 

f 


Section I 

Electrical Parameters of Crystal Units 

activity, the latter probably being due to higher 
motional resistances for the unwanted modes. Un- 
wanted modes are not always accompanied by 
changes in the resistance. For example, a sudden 
jump from one frequency to another, but without 
the dipping effects shown in figure 1-114, where 
the temperature-frequency curve is effectively 
broken into two smooth curves, may have very 
little effect on the activity. This type of frequency 
jump, which was quite common in the old Y-cut 
crystals, seems to be due primarily to small de- 
fects in the finishing of the crystal blank. Where 
only one such jump occurs during the temperature 
cycle, it can usually be eliminated by a slight re- 
tuning of the oscillator circuit. However, retuning 
the oscillator circuit, particularly if the crystal is 



CSVSTAI. TEMPERATURE , CENTIGRADE DEGREES 

Figure 1-113. large clockwise angle of rotation of frequency-temperature curve of harmonic-mode A element 
caused by small increments (3 minutes of arc) in the cutting orientation angle about the X axis 


WAOC TR 56-156 


107 



Section I 

Electrical Parameters of Crystal Units 

being operated near series resonance, will have 
little effect upon those temperature-frequency 
characteristics due to unwanted modes that are in- 
herent functions of the major dimensions of the 
crystal blank. A crystal unit having characteristics 
similar to those shown in figure 1-114 should not 
be used where the operating temperature is ex- 
pected to extend into the erratic region. Unfor- 
tunately, the specifications for most of the crystal 


units listed in Section II of this handbook are not 
rigorous enough to provide a guarantee against 
unwanted modes for every type of unit, if the 
effects upon the frequency and the effective resist- 
ance do not cause over-all deviations beyond the 
maximum allowed. On the other hand, “jumpy” 
crystals are the exception rather than the rule, 
but if particular precautions are necessary where 
wide temperature variations are to be encountered. 




. 

VCTIVITY-l 

EMPERATl 

JRE CHAR/ 

^CTERISTK 






"Ny — ' 





r 



Y 







L 














t 



CRYSTAL TEMPERATURE, CENTIGRADE DEGREES 


Figure 1-114. Activity-temperature and frequency-temperature characteristics of harmonic-mode A element, 

showing effects of unwanted modes 


WADC TR 56-156 


108 




i>^liilfi:«liose crystal units should be used which are 
specified by Military Standards to be free of un- 
'^an^d modes over the desired temperature range. 

1-255. The overtone modes of the thickness-shear 
eleihei)^ are more likely to be troubled with spuri- 
ous frequency dips of the type shown in figure 
1-114 th^ are the fundamental modes, but a crys- 
tal tiiat is erratic at its fundamental vibration 
usually exhibits a pure frequency spectrum at a 
high harmonic. Indeed, because the frequencies of 
the unwanted and the desired harmonics do not 
increase in the same proportion, one method of 
lessening the probability of interfering modes at 
the higher harmonics is to deliberately cut the 
crystal with edge dimensions which favor spurious 
responses at the fundamental frequency. Never- 
theless, the overtone crystals have a tendency to 
oscillate at two or more thickness-shear frequen- 
cies. Usually, this seems to be due to slight differ- 
ences in the thickness of the crystal from one point 
to another. For each order of the harmonic, n, the 
crystal can be imagined to be divided into n layers 
perpendicular to the thickness, with each layer be- 
ing a separate crystal vibrating 180° out of phase 
with the neighboring layers on each side. If n is 
an even number, the separate sections tend to can- 
cel each other’s electrical effects at resonance. For 
this reason the even harmonics cannot easily be 
electrically excited. In the case of the odd har- 
monics, there is always effectively one vibrating 
layer whose alternating polarity is not neutralized. 
Most pf the activity is more or less centered in 
one particular region of the crystal plate. If the 
thickness at an active point differs slightly from 
the thickness at a neighboring point, there will be 
a tendency to jump from one activity center to 
another, and small jumps in the frequency can 
result. In the case of crystal plates vibrating at 
high harmonic modes, a small variation in the 
thickness dimension is generally more likely to 
produce a sudden frequency jump than if the same 
crystal were vibrating at its fundamental mode. 
If there is little difference between the equilibrium 
conditions of two vibrating stages, the frequency 
may shift back and forth at an audio rate, thereby 
effectively modulating the oscillator output with 
an audio frequency. Such frequency jumps are best 
avoided by the use of ceramic-button holders, the 
design of which concentrates the excitation in a 
small area at the center of the crystal where the 
most uniform thickness is attainable. Occasion- 
ally, it is found that the small frequency jump 
occurs only at a particular adjustment of the oscil- 
lator, and therefore it can be avoided by slight 
changes in the oscillator tuning. Even so, unless 


Section I 

Electrical Paramotere of Cryttal Unit* 

the temperature is to remain reasonably constant, 
a crystal unit exhibiting any tuning jump at all 
should not be used. For although an unwanted 
mode that occurs during a temperature cycle may 
never appear during a tuning adjustment, the re- 
verse situation is rarely found — a frequency jump 
that can be caused by a tuning adjustment is al- 
most certain to appear during a temperature cycle. 

EFFECT OF DRIVE LEVEL UPON 
CRYSTAL PARAMETERS 

1-256. There is insufficient data and standardiza- 
tion at the present time to analyze or to predict 
exactly the effect a change in the drive level will 
have on a crystal unit of a given type. Not only 
do crystal units of the same type exhibit various 
reactions, depending on the nominal frequency, 
the method of fabrication, and the manufacturer’s 
specifications, but even when all these factors are 
the same for a sample of crystal units, the indi- 
vidual reactions to changes in drive level are un- 
predictable. The frequency and series-arm resist- 
ance curves versus drive level in figures 1-115 and 
1-116 are shown as examples. These curves were 
prepared from data obtained during a Signal 
Corps research project at New York University 
by a research team consisting of Messrs. Don J. R. 
Stock (Director), L. Silver, E. Strongin, A Yev- 
love, and A. Abajian. The curves in both figures 
were made from the same set of 9-mc crystal units 
— AT-cut, electrode-plated, wire-mounted types 
CR-18/U and CR-19/U, all made by the same 
manufacturer. 

FREQUENCY VERSUS DRIVE 

1-257. In figure 1-115, note that although the fre- 
quency of the average crystal unit tends to in- 
crease with drive level, this effect is by no means 
to be found at all drive levels for all crystal units. 
Unfortunately, the temperature-frequency curves 
for these same crystals are not available, so it is 
not possible to judge how much of the frequency 
deviation is due simply to the rise in temperature 
with drive level. However, the frequencies of other 
A elements have been tested for frequency devia- 
tion versus power, and even though the increases 
of temperature due to drive occur at points 
of negative slope on the frequency-temperature 
curve, the actual frequency-drive level curve gen- 
erally reveals a positive slope. This increase in 
frequency with drive is apparently due to a rela- 
tively large temperature-gradient coefficient. The 
net effect on the frequency is due to the combined 
influences of the changes in both the temperature 
and the temperature gradient, which influences 


WADC TR 56-156 


109 



Section I 

Electrical Parameters of Crystal Units 



Figure 1-115. Frequency deviation versus drive for a random sample of 9-me A elements. All crystal units are 
the products of the same manufacturer and are similarly fabricated and mounted In HC-d/U holders 



POWER IN MILLIWATTS 

Figure 1-116. Resistance deviation versus drive for a random sample of 9-mc A elements. Curves are for the 
same sample of crystal units whose frequency-drive characteristics are shown in figure 1-115. Correspondingly 

numbered curves are those of the same crystal unit 


WADC TR 56-156 


110 



Section I 

Electrical Parameters of Crystal Units 


may or may not be in opposition. From the appear- 
ance of the curves in figure 1-115, it is possible 
that those curves starting with a negative slope 
may be primarily responding according to the nor- 
mal temperature coefficient. No data is available 
concerning the degree by which the orientation of 
the crystal plate relative to the mounting wires 
might influence the thermal-gradient effect. 

1-258. The rise in temperature per milliwatt of 
drive varies widely with the types of mounting 
used and the sizes of the crystal plates. For wire- 
mounted units, most of the heat generated is due 
to friction at the points where the crystal is sup- 
ported. With the heat source thus concentrated in 
a small region of the crystal surface, steep thermal 
gradients can be expected. The over-all rise in 
temperature is also greater in the case of wire- 
mounted units, since most of the thermal-leakage 
must be through the air, which, like all gasses, 
acts as a thermal insulator. If the crystal unit is 
vacuum-sealed, the temperature change per milli- 
watt may increase by a factor of from two to ten, 
depending upon the size of the supporting wires 
and how much of the crystal surface is metal- 
plated. With the air evacuated, the heat leakage is 
primarily through the supports and by radiation. 
The amount lost by radiation depends largely 
upon the emissivity of the crystal surface, which 
is approximately 40 times as great for unplated as 
for plated areas. If it can be assumed that the heat 
is evenly distributed over the volume of the crystal, 
the temperature rise of a one-centimeter-square 
crystal wire-mounted in an HC-6/U holder (not 
evacuated) can be expected to be approximately 
0.3 to 0.4 centigrade degrees per milliwatt of 
drive. In practice, however, the temperature of the 
parts of the crystal where most of the heat is gen- 
erated may increase as much as 10 times this 
amount. If high or variable drive levels are to be 
used, pressure-mounted crystal units should be 
employed. The relatively large contact area be- 
tween the crystal and the supporting electrodes 
permits a more uniform distribution of the heat, 
thereby reducing the magnitudes of the thermal 
gradients. The pressure mounts also provide a 
much higher thermal conductivity away from the 
crystal, thus enabling a much smaller temperature 
rise per milliwatt of drive. Finally, the pressure 
mount provides better mechanical and aging pro- 
tection for the crystal when operated at high- 
amplitude vibrations. Regardless of the type of 
mounting, it is never desirable from the point of 
view of stability or of long crystal life to use a 
higher crystal drive than absolutely necessary. 
1-259. In tests made with GT-cut crystals, where 



MICROAMPERES PER MM OF WIDTH 


Figure 1-117. Frequency deviation versus crystal 
current density for two GT-cut crystals which 
were sub/eeted to different periods of etching* 

the frequency deviation with temperature is prac- 
tically zero over a 100-degree centigrade range, 
the opportunity has been afforded to study the 
frequency deviation due to drive alone without the 
complication of temperature-coefficient effects. Ex- 
periments with G elements, as reported by R. 
D’Heedene, reveal a negative frequency deviation 
with drive, as shown in figure 1-117. Note that the 
GT plate given a deep etch maintained its stability 
during much higher drive levels than did the plate 
etched only 20 minutes. Since the effective resist- 
ance of the better finished crystal can be expected 
to be less than, and to be more stable with increas- 
ing drive than, the resistance of crystal A, the 
changes in frequency with changes in crystal 
power may have been much closer than the curves 
in figure 1-117 indicate if it was assumed that the 
resistances of the two crystals were equal. If the 
change in frequency is due primarily to changes in 
the thermal gradients, it is more directly a function 
of the crystal power. On the other hand, if the fre- 
quency deviation is due primarily to mechanical 
strains resulting from high amplitudes of vibra- 
tions, it is more directly a function of the crystal 
current. Although the evidence now suggests that 
it is the thermal gradients that are the primary 
factors, certainly a lowering of the frequency can 
be expected for any mode if the elastic limit is 
approached too closely. After crystal units are 
subjected to high amplitudes of vibration, they do 
not return immediately to their original frequen- 
cies when the drive is reduced to a low level. A 
period of days or weeks may ensue before the 
crystal unit regains its former characteristics, 
during which time the performance resembles that 
of a crystal rapidly aging. 

1-260. Although the frequency-versus-drive char- 
acteristics of individual crystal units deviate con- 
siderably from the norm, the characteristics are 
generally similar enough to plot reasonably de- 


WADC TR 56-156 


111 




Section I 

Electrical Parameters of Crystal Units 

pendable average curves when the fabrication 
processes and the frequencies are the same. Such 
curves showing average frequency deviation versus 
power are illustrated in figure 1-118. Each curve 
represents the average of several samples from 
a representative manufacturer for a given fre- 
quency. The curves with the same letter corre- 
spond to crystal units of the same manufacturer. 
All the crystals are A elements, metal-plated and 
wire-mounted in HC-6/U holders. In every case, 
it can be seen that the average tendency is for the 
frequency to increase with power. 

RESISTANCE VERSUS DRIVE 

: -261. The resistance curves shown in figure 1-116 
are more or less typical of the wide variations that 
must be considered in the design of an oscillator. 


A minimum performance level must be maintained 
regardless of the resistance of the crystal unit, as 
long as the resistance complies with the military 
specifications. Actually, the average series-arm 
resistance of the crystal units shown is quite low 
for 9-mc crystal units. As would be expected, the 
resistance generally increases as the amplitude of 
vibration increases. About one crystal unit in 
eight, however, exhibits a steady decrease in re- 
sistance as the drive increases. The initial resist- 
ance of such a unit is usually higher than the 
average. Note in figure 1-116 that a number of 
the curves have relatively sharp negative slopes at 
very low power levels. This characteristic is not 
uncommon, particularly in the case of harmonic- 
mode elements, where it has become a problem 
requiring special test procedures. Harmonic-mode 





POWER IN MILLIWATTS 


Figure 1-7 18. Average frequeney deviation versus drive. Bach curve represents the average of a random sample 
of severai simiiarly constructed units of one manufacturer. Curves having the same letter represent the character- 
istics of crystal units of the same manufacturer 


WADC TR 56-156 


112 









crystel units are now required to pass performance 
tests at two drive levels. The first is at the normal 
maximum recommended drive level; the second 
is to ensure that the resistance falls within 
specifications when the drive is at a minimum. In 
the jcase of fundamental thickness-shear elements, 
sharp negative slopes of the resistance-drive curves 
at low drive levels are not as common an occur- 
rence percentage-wise as is suggested by the 9-mc 
saipples in figure 1-116. Much more likely to be 
found are resistance curves with the slopes slightly 


SecKon I 
Crystal Oscillators 

more positive at very low drive levels. 

1-262. In the design of an oscillator for military 
equipment a principal consideration is to ensure 
that the crystal drive does not exceed the recom- 
mended maximum when one crystal unit of the 
same standard type, but of perhaps a greatly 
different resistance, replaces another. If the drive 
is not kept to the lowest practicable level, the re- 
sistance of a borderline crystal may well be in- 
creased beyond the permissible limits, thereby 
excessively degrading the oscillator stability. 


CRYSTAL OSCILLATORS 

For a comprehensive cross-index of crystal- 
oscillator subjects, see end of Section /. 


FUNDAMENTAL PRINCIPLES OF 
OSCILLATORS 

1-263. An oscillator can be defined as any physical 
system having a periodic motion. If the motion is 
plotted as a function of time, a wave shape or a 
sequence of wave shapes that fairly accurately 
repeats itself would be considered the fundamental 
cycle of a stable oscillator. On the other hand, if 
there were a continuous change in the wave shape, 
the oscillator would be classified as being unstable. 
An oscillator that is unstable in the general sense, 
may, however, have a stable component of fre- 
quency, or amplitude, or some combination thereof. 
Of course, all oscillating systems are unstable to 
some degree, so that the terms stable and unstable 
define classifications that are somewhat arbitrary 
though none the less convenient. 

1-264. Oscillators may also be classified according 
to the way in which the oscillations are controlled. 
A number of classifications are possible, but of 
those which consider the oscillating system alone, 
there are three general types: free, forced, and 
forced-free. Free oscillators are those whose oscil- 
lating energy is entirely self-contained in the 
oscillating state, and whose waveform and fre- 
quency are determined entirely by the properties 
of the system. The solar system, purely from the 
point of view of the planetary motions, is an 
example. A quartz crystal vibrating freely in space 
is another. Forced oscillators are those in which 
the energy, wave shape, and frequency are under 
the control of an external power source. An ex- 
ample would be the cone of a loudspeaker, or a 
quartz filter crystal, where the vibrations are con- 
trolled by the signal source. “Forced-free” oscil- 


lators can be described as those which are driven 
by the energy of an external source, but where the 
frequency is primarily determined by the prop- 
erties of the system. Crystal oscillator circuits are 
of the forced-free type. Again, the classification 
is somewhat arbitrary, for in the final analysis 
there are no absolutely free nor absolutely forced 
oscillations, nor can two systems be rigorously 
considered as distinct when there is an exchange 
of energy between them. In fact, fourth and fifth 
categories are possible. In the one, the frequency 
control and drive are both inherent in the system, 
yet not in the same sense as that defined for free 
oscillators. By a stretch of the imagination, a good 
example is to be found in the hula dancer. In a 
fifth category, the energy is supplied by the oscil- 
lating system, but the frequency is controlled ex- 
ternally. An example is to be found by considering 
each limb of the hula dancer as a separate oscil- 
lating system. Still other categories are possible. 
Insofar as a crystal oscillator circuit is concerned, 
as distinct from its power source, we can consider 
it an independent controlling system in respect to 
the frequency, but only to the extent that the cir- 
cuit can predetermine the periodic characteristics 
of both the input and the output energies. 

FUNDAMENTAL REQUIREMENTS OF STABLE 
FORCED-FREE OSCILLATIONS 

1-265. There are two fundamental conditions that 
are always met when a physical system is being 
maintained in a stable state of forced-free oscil- 
lation. First, the primary source of energy, or 
“prime mover,” is supplying energy at the same 
average rate at which energy is being expended 


WADC TR 56-156 


113 


Section I 

Crystal Oscillators 

by the system. Second, all forces acting on the 
oscillations are, themselves, stable periodic func- 
tions which have frequencies relative to the fre- 
quency of the oscillator that can be expressed by 
f 18 

rational numbers (e.g., = 1 . 2 » 7 ’ 1 *' 

practical case, this latter property cannot occur 
simply by coincidence between independent sys- 
tems, so that the condition implies that the periods 
of all forces acting on the stable oscillations are 
controlled by the oscillator, itself. The most im- 
portant of these forces are those exerted by the 
power source in driving the oscillator and those 
exerted by the output as a result of reaction with 
the load. The first condition for stable oscillations 
ensures that the average amplitude of oscillation 
is stable. The second condition ensures that the 
fundamental frequency is stable. Together, they 
ensure that the waveform is stable. 

1-266. As a simple example of a forced-free oscil- 
lator, consider a system consisting of a swinging 
pendulum. If the frictional losses per cycle are 
small relative to the energy stored in the system, 
and if the amplitude is small, the oscillations are 
essentially those of simple harmonic motion, with 
the frequency being determined by the geometry 
of the system and the gravitational field. Assume 
that the pendulum, each time it reaches a certain 
point in its cycle, triggers a latch that releases a 
spurt of energy from a power source. If each kick 
received imparts the same amount of energy to the 
system, an equilibrium of stable oscillations will 
be reached when the input pulses are being com- 
pletely transformed into a simple harmonic flow 
of energy to the surroundings. In order for the 
power source to transmit energy to the system, it 
must exert its force while the system is moving in 
the direction of the force. Otherwise, it will be 
the “power source,” rather than the oscillator, that 
gains energy. If the system dissipates energy at 
the exact instantaneous rate at which it is received, 
the applied force will not, itself, produce momen- 
tary accelerations in the pendulum’s swing each 
cycle. However, the losses from the system do not 
occur at simply a single interval during the cycle, 
but obey a sine-wave function extending over the 
entire period. Only at the instants of zero kinetic 
energy, at the end of each swing, can the instanta- 
neous losses be considered zero. Thus, each pulse 
of energy must accelerate the pendulum in its di- 
rection of motion so that the waveform must 
deviate somewhat from a pure sine shape. The dis- 
tortion is a minimum when the ratio of the total 
stored energy to the energy input per cycle is a 


maximum, and when, during the input interval, 
the ratio of the energy dissipated to the energy 
absorbed is a maximum. In other words, the dis- 
tortion becomes smaller the higher the “Q” of the 
pendulum, the longer the interval over which the 
impulse is spread, and the more the impulse is 
centered at the middle of the swing where the in- 
stantaneous power dissipation is the greatest. 
Now, even though the impulses distort the wave- 
form, the oscillations are stable if exactly the same 
pattern is repeated periodically. If instead of 
swinging back and forth, the pendulum is swing- 
ing through a complete circle, it is easier to see 
that if several impulses are transmitted to the 
system each cycle, the same pattern will continue 
to be repeated as long as the frequency, or fre- 
quencies, of the impulses are related to the 
frequency of the pendulum by a ratio of whole 
numbers. The fundamental of the pendulum cycle 
need not equal the fundamental of the stable wave, 
but it must be a harmonic thereof. For example, 
suppose that a pulse of energy is imparted to the 
system only once every four cycles. Then the funda- 
mental period of the stable waveform is four times 
the period of the pendulum. If five impulses are 
delivered for each four cycles of the system before 
being repeated in the same phase as before, the 
period of the stable waveform will again equal 
four pendulum cycles. Only when there are one, 
two, three, etc, impulses repeated each cycle does 
the period of the stable wave equal the natural 
period of the oscillator. In the same way by which 
the forces exerted by the energy sources distort 
the sine-waveform, so do the forces exerted by the 
load into which the pendulum loses its energy. If 
the impedance during any interval of the swing 
changes from cycle to cycle without repetition, a 
stable waveform is not obtainable. 

APPLICATION OF FUNDAMENTAL OSCILLATOR 
PRINCIPLES IN THE DESIGN OF 
ELECTRONIC OSCILLATORS 

1-267. In the application of electronic oscillators, 
it is not usually a stable over-all waveform that is 
the first requirement, but a stable fundamental 
frequency. Practically, however, these two effects 
are not independent, and the generation of the one 
involves the generation of the other. *1116 deviation 
from a pure sine wave in the a-c output of a stable 
crystal oscillator will be entirely due to the pres- 
ence of harmonics of the fundamental. Such frac- 
5 

tional components as -r ths of the fundamental 
4 

or the harmonics thereof do not appear. With the 


WADC TR 56-156 


114 



load impedance constant, the conditions of stability 
are reached when energy is being supplied at the 
average rate of dissipation and at the same phase 
interval of each cycle. In the generalized crystal 
oscillator circuit of figure 1-108 (B), the first con- 
dition is met when p, is equal to — R,. The second 
condition is met when X, is equal to —X,. In ap- 
plication, the two conditions are not independent 
of each other, for the build up in the energy of 
oscillation depends upon the power source not 
exerting its force in phase opposition to the oscil- 
lations. Indeed a common approach to the analysis 
of an oscillator circuit is to establish a single equa- 
tion that expresses simultaneously the equilibrium 
requirements of both the rate and the phase of the 
energy supply. Equation 1 — 289 (1) for the Pierce 
and Miller circuits is an example. Since this type 
of equation, when fully developed, usually becomes 
quite cumbersome, such an approach is only oc- 
casionally followed in this manual, it being more 
convenient to treat the two basic equilibrium con- 
ditions separately. The first condition, that the 
rates of energy supply and of energy dissipation 
be equal, can be assumed to be satisfied if the mag- 
nitude of the rms voltage between any two points 
in the circuit is constant. This condition can be 
expressed by an equation which equates the loop 
gain to unity. By this we mean that, starting with 
the input circuit, or at any convenient point, the 
overall voltage gain around the oscillator loop back 
to the starting point is unity at equilibrium. If the 
loop gain is greater than unity, oscillations build 
up, if less than unity, they do not start, or, if 
already started, they die down. As an example, the 
loop-gain equation of a simple oscillator of the type 
shown in figure 1-177 (D), where the feedback 
energy is transformer-coupled from the plate cir- 
cuit to the grid circuit, can be expressed as follows ; 

G,G.G3 =1^^ • • |£. = 1 1—267 (1) 

where G, is the gain of the vacuum tube, Gz is the 
gain of the transformer in the plate circuit, and 
G:t is the gain of the feedback from the transformer 
secondary to the vacuum-tube input. Although the 
loop-gain equation may at first glance appear 
trivial since the product of the voltage ratios 
equals unity regardless of what voltage values are 
assigned, it should be remembered that a necessary 
qualitative implication requires that each voltage 
ratio represent the gain of an actual transfer of 
energy from one circuit to another. When the 
voltage ratios are expressed in terms of the circuit 
parameters an overall network formula is estab- 
lished that will serve to discipline the oscillator 


Section I 
Crystal Oscillators 

design. In a similar manner, the second condition 
of equilibrium can be expressed as an equation of 
the loop phase rotation, in which the total phase 
shift in the voltage around an oscillator loop is 
equal to zero, or to some integral multiple of 360 
degnrees. Continuing the example of the trans- 
former-coupled oscillator above : 

+ 6.P -f Ogs = 0 (or 360°) 1—267 (2) 

where is the phase of E,, with respect to Eg, 
is the phase of E„ with respect to E,„ and is the 
phase of Eg with respect to E,. In most cases ap- 
proximately ideal conditions can be assumed so 
that the loop phase requirements need only be 
analyzed qualitatively. For instance, in the ex- 
ample given of the simple transformer-coupled 
oscillator, let it be imagined that the plate circuit 
is to be designed so that the vacuum tube faces a 
purely resistive load. Thus, 6,,^ will represent the 
180-degree phase reversal introduced by the vac- 
uum tube, fl„|, will represent a counter 180-degree 
reversal by the plate transformer, so that the 
principal phase consideration is to design a grid 
circuit that will allow Eg to be in phase with E. at 
the desired frequency. The loop-phase considera- 
tions are much more involved in the case of the 
conventional one-tube resonator circuit shown in 
figure 1-119. The loop phase rotation in this gen- 
eral type of circuit applies to such oscillators as 
the tuned-grid-tuned-plate, the Hartley, the Col- 
pitts, the Pierce, and the Miller. It is discussed in 
detail in following paragraphs. The loop equations 
are the guides by which the design engineer ap- 
proaches the basic oscillator problems of obtaining 
the desired amplitude, the desired frequency, the 
desired amplitude stability, and the desired fre- 
quency stability. As a general rule, these four fun- 
damental design considerations are handled with 
the aid of the loop equations in the following ways : 

a. An oscillator is designed to provide a certain 
amplitude of oscillation by ensuring that the 
parameters that vary with the amplitude (usually 
the vacuum-tube parameters) reach their limiting 
values, as defined by the loop-gain equation, when 
the desired amplitude is reached. 

b. An oscillator is designed to oscillate at a 
given frequency by ensuring that the loop-phase 
equation holds at, and usually, only at, the desired 
frequency. Should the loop-phase equation also 
have a solution at some other frequency (e.g. the 
loop phase of the transformer-coupled oscillator 
mentioned above may well equal zero at more than 
one mode of the crystal’s vibration), the design 
must ensure that the loop gain is less than unity at 
the unwanted frequency. 


WADC TR 56-156 


115 



Section I 

Crystal Oscillators 

c. The amplitude stability is improved by coun- 
teracting or minimizing variations in those circuit 
parameters which, as indicated by the loop-gain 
equation, are most likely to cause changes in the 
amplitude. 

d. The frequency stability is improved by coun- 
teracting or minimizing variations in those circuit 
parameters which, as indicated by the loop-phase 
equation, are most likely to cause changes in the 
frequency. 

Before proceeding to a discussion of the particular 
types of oscillators, let us first examine in detail 
the phase relations of the conventional resonator 
circuit of figure 1-119 (B). If a firm qualitative 
understanding of the operation of this type of cir- 
cuit is had, the reader should be greatly aided in 
interpreting the physical meaning of equations 
later to be derived. 

PHASE ROTATION IN VACUUM-TUBE 
OSCILLATORS 

1-268. The conventional equivalent circuit of a 
vacuum-tube amplifier is shown in figure 1-119 
(A). The equivalent generator voltage is equal to 
— where n is the amplification factor of the 
tube, and Eg is the excitation voltage on the grid. 
Rp is the plate resistance of the tube, and Zi, is 
the a-c load impedance. The minus sign of the 
generator voltage indicates a 180-degree phase 
difference between the equivalent emf and Eg. For 
oscillations to build up, energy must be fed back 
in the proper phase from the plate circuit, or from 
some circuit in a following stage. The control grid 
of the vacuum tube is effectively an escapement 
device for controlling the release of energy from 
the power source. Since this energy must be re- 


leased each cycle so as not to be in phase opposition 
to the oscillator, the grid voltage alternations must 
be “timed” by the activity in the rest of the circuit. 
This means that a sufficient and properly phased 
part of the energy released by the action of the 
grid must be fed back from the plate circuit each 
cycle, or from some other circuit of a following 
stage, in order to continue the periodic release of 
energy. The initial rush of plate current is to be 
sufficient to shock the circuit into oscillation, and 
the initial alternating voltage fed back to the grid 
circuit must be sufficient for the vacuum tube to 
generate more a-c energy than is lost during the 
first cycle. Rp increases with the amplitude of 
oscillations until equilibrium is reached. 

1-269. The phase relation between the grid and 
plate voltages of an oscillator vacuum tube at equi- 
librium is the same as that which would occur if 
the grid were excited at the same frequency from 
an external a-c source and the tube were connected 
as a conventional amplifier, operating into the same 
equivalent load impedance it faces as an oscillator. 
However, only a certain impedance relationship 
among the components of a particular oscillator 
circuit can provide a feed-back producing the 
proper input phase. It is this necessary impedance 
relationship that determines the frequency. In the 
usual single-tube oscillator, the equivalent vacuum- 
tube generator, of voltage — /»Eg, must drive a 
plate-coupled feed-back circuit that causes the volt- 
age appearing across the grid to be rotated 180 
degrees ahead of or behind the generator emf. The 
simplest method of reversing the phase is by trans- 
former coupling. On the other hand, if two tubes 
are used, the reversal can be accomplished by the 
second tube alone. Either of these methods can 




figure 1-119. 


(A) Equivalent circuit of vacuum-tube amplifier. (B) Equivalent circuit of crystal oscillators of the 

Pierce and Miller types 


WADC TR 56-156 


116 


enable a crystal oscillator to work into a more or 
less resistive load, so that fluctuations in the circuit 
parameters can have little effect on the feed-back 
phase, and, hence, upon the frequency. In the con- 
ventional parallel-resonant circuits, such as are 
illustrated in figure 1-109, the phase is rotated as 
shown in figure 1-119 (C). First, assume an ideal 
case in which the resistive losses in the feed-back 
arm are zero. In this case, but only in this case, 
Zl would need to be resistive. The frequency would 
be that at which the plate and feed-back arms 
operate as a parallel-resonant tank. There would 
be no phase shift in the voltage across Zl, and Ep 
would be of the same sign as — /lE,. Zp and Z„ the 
impedances of the plate circuit from plate to cath- 
ode and of the grid circuit from grid to cathode, 
respectively, are reactive, and must always have 
the same sign. Zpg, the plate-to-grid impedance is 
the dominant impedance in the feed-back circuit, 
and is always opposite in sign to Zp and Zg. In the 
ideal circuit, if Zp and Z, are positive, Zp, is nega- 
tive, so the current, I,, leads Ep, and therefore 
—nEg, by 90 degrees. If Zp and Z, are negative, Zp, 
is positive, so I, lags Ep 90 degrees. The voltage 
across Zp,, of course, would be in phase with Ep in 
both instances. Since Z, is opposite in sign to Zp„ 
Eg thus is opposite in sign to Ep, and the required 
reversal takes place. Note that I, is first rotated in 
phase with respect to Ep; next E, is rotated in the 
same direction with respect to I,. 

1-270. In an actual circuit, the feed-back losses 
cannot be zero, so that a 180-degree reversal can- 
not be obtained in the conventional feed-back cir- 
cuit alone. This means that Ep must first be rotated 
by an amount exactly sufficient to make up the 
difference. Assume first that Rp is much greater 
than Zl, so that Ip can be assumed to be essentially 
in phase with the equivalent generator voltage. In 


Section I 
Crystal Oscillators 

this case, an inductive Zl causes Ep to lead the emf, 
whereas a capacitive Zl causes Ep to lag the emf. 
Unless the effective Q of the feed-back circuit is 
very low, Zl, must be very nearly resistive, for the 
shift in the phase of Ep need not be large. In any 
event, the rotation of Ep must be in the same direc- 
tion as that of I, and E,. For this to occur, the 
susceptance of Zp must be greater in magnitude 
than the susceptance of (Zp, + Z,). That is, the 
reactive component of the current through Zp must 
more than cancel the reactive component of I,. The 
smaller the value of Rp compared with the value 
of Zl, the more nearly will Zl control the phase of 
Ip, and the more detuned must the parallel circuit 
become in order to obtain the necessary rotation 
of Ep. If practically all the resistance in the feed- 
back arm is between the grid and the cathode, as 
is normally the case when E, is developed directly 
across the crystal unit, Ep must be rotated through 
a larger angle than otherwise, thereby requiring 
the parallel circuit to be detuned to a greater de- 
gree. This is because Ep must be rotated by an 
amount effectively equal to the sum of two angles. 
One of the angles is the difference between the 
actual phase of I, relative to Ep and the ideal phase 
of ±90 degrees. The second angle is the difference 
between the actual phase of Z, and its ideal phase 
of ±90 degrees. If all the resistance is contained 
in the large impedance Zp,, only the phase de- 
ficiency of I, is reflected in the phase of Ep. On the 
other hand, if all the resistance is effectively con- 
tained in the small impedance Z„ the effect on the 
phase of Ep by I, is normally small by comparison 
with the effect due to the grid-to-cathode resist- 
ance. Expressed in polar form : 

— IgZg (^i« + ^zg) “ IgZg (®°) 
where fii, and dzg, which must be equal in magnitude 


characteristics of eOUIVALEHT-CIRCUIT PARAMETERS 

i90* 

PHASE RELATIONS 180«^ — 

t-90* 

{VOLTAGE AND CURRENT WITH RESPECT TO E g. 1 MPE DANCES 
with respect TO CURRENT THROUGH THEM ) 

z P 

^9 

Z pg 

BO 

jzp/Zpgcj 


Zl 

ip 

E P 


1 g 

Z g 

E g 



-Ht- 

<OR > ‘ 

&3 i 





n 

♦ 








1# 




» 1 





♦ 

♦ 

♦ 









-nnnp — vw- 



» 1 

» 0.95 


/ 


/ 


\ 




If 

-onnp' — 

>> 1 

as 0.99 


' — 





♦ 



1 m 





•vvvA 

-TTTinr' — 

If 

S3 > 

< 0.95 




/ 


\ 



II 




MM ... 

I# 


as 1 

< 0 95 






♦ 

* 



II 





( C I 


Figure I- 1 19. (C) Chart showing ideal and typical phase relations necessary for forced-free 

oscillations of the circuit shown in (B) 


WADC TR 56-156 


117 




















Section I 

Crystal Oscillators 

but opposite in sign, are the phase angles of I, with 
respect to Eg and of Z* with respect to I*. Now, 
letting dzvsc equal the phase of the total feed-back 
impedance, (Zpg + Z,) , with respect to the current 
through it, and 5 ep equal to the phase of Ep with 
respect to Eg, 

we have ^Kg = = 0 

and ^Kp ^ig ^Zpgc 

so that = Ozpgc — 0z^ 1 — 270 (1) 

Since flzpg,. is opposite in sign to 0zg, these two phase 
angles add numerically. If it is assumed that Zpgc is 
approximately six times the magnitude of Zg, but 
that all the feedback-arm resistance is between 
the grid and cathode, the Q of Zpgc will be approxi- 
mately five times the Q of Zg. Under these condi- 
tions the rotation of Ep from the ideal value of 
—180 degrees is approximately 20 per cent greater 
than the deviation of —Bzg from its ideal value of 
3:90 degrees. If Zg represents the effective imped- 
ance of a crystal unit in parallel with the grid-to- 
cathode capacitance and resistance, the minimum 
rotation of Sep occurs when the effective Q of the 
crystal and its shunt impedance is a maximum, 
provided Zpg > > Zg. Similarly, if Zpg represents the 
Zc of a crystal unit whose effective resistance is 
much greater than the equivalent series resistance 
of the grid-to-cathode impedance, the rotation of 
Sep depends primarily upon the crystal unit Q., and 
is a minimum when Q, is a maximum. 

1-271. In a conventional parallel-resonant crystal 
oscillator having an ideal feed-back arm, the fre- 
quency would be determined entirely by the reson- 
ance of the tank circuit, so that fluctuations in Rp, 
although effective in changing the activity, would 
not affect the frequency. In practice, the equivalent 
resistance of the crystal unit is a parameter of the 
feed-back arm, so that the detuning of the tank 
becomes very nearly a direct function of the effec- 
tive Q of the crystal and its shunt capacitance. 
Note that the phase of Ep with respect to E* and 
— /lEg is determined entirely by the parameters of 
the feed-back arm. As long as oscillations continue, 
variations in Rp or Zp can only change the phase of 
Ep indirectly, i.e., by causing a change in the Q of 
the feed-back arm. Since the effective resistance 
can vary by as much as a factor of 10 between 
minimum and maximum values for the same stand- 
ard type of crystal unit, an oscillator cannot be 
designed too closely upon the assumption that the 
load impedance Z,. will be essentially the same 
either in phase or magnitude when one crystal unit 


is replaced by another — even if it is of the same 
type and nominal frequency. This limitation might 
possibly be minimized by switching the crystal 
unit from the feed-back circuit to the plate circuit, 
and replacing it with a high-Q inductor in the 
feed-back arm. If the grid were operated with bias 
sufficient to prevent the flow of grid current, a very 
high and predictable Q could be obtained in the 
feed-back circuit, and an approximately resistive 
Zi, could be assumed for the tank regardless of the 
variations in Zp due to variations from one crystal 
unit to the next. 

1-272. An oscillator that can be represented by the 
equivalent circuit shown in figure 1-119 will show 
the following phase characteristics and related 
effects. 

a. The phase of Ep is entirely determined by the 
over-all Q of the feed-back circuit and the Q of the 
grid-to-cathode impedance, Zg. The phase angle, 
Sep, is given by equation 1 — 270 (1) as being equal 
to (tfzpgc —0zg)- Let us now assume that (flzi«c 
—6ze) is determined by an imaginary over-all Q 
of the feed-back circuit. This we define to be 

Qr = I cot Sep 1 = I cot (0zpgc - 6zg) \ 

^ tan ^zpgc tan dzg 4- 1 
tan dzpgc - tan dzg 

_ Qpkc Qk ~ 1 _ Qi'gc Qk 1—272 

Qp« + Qg Qpgc + Qg ^ ’ 

It can be seen that if either Qpg,., the actual effec- 
tive over-all Q of the feed-back circuit, or if Qg, 
the Q of the grid-to-cathode impedance, is very 
large compared with the other, Qf is approximately 
equal to the smaller Q. 

b. For a given phase difference between Ep and 
— /lEg, the ratio of Zp to the total feed-back imped- 
ance, Zpgc, is less than 1 by an amount which in- 
creases as the ratio of Rp/Zt decreases. In other 
words, the ratio of the r-f current in the plate cir- 
cuit to the r-f current in the grid circuit increases 
as Rp decreases. 

c. The value of Rp/Zi, is partly a function of the 

2 

ratio = Ep/Eg. Assume, for example, that 

Rp > > Ze, so that Ep/^Eg (= |^. Also 

assume that during oscillations Zg is decreased, but 
that Zpg remains essentially constant. The ratio 
Ep/Eg is thus increased, and likewise the ratio 
2 

Ep//iEg In other words, as Eg becomes a 

Kp 

smaller component of the total voltage across the 


WAOC TR 56-156 


118 



feed-back circuit, Ep cannot decrease in the same 
proportion, else each succeeding cycle would be 
weaker than the one before; so the ratio Rp/Zl 
must decrease. Part of the change is due to the 
increase in Z,., and part is due to a decrease in Rp. 
If it is assumed that a large percentage change in 
Z„ causes only a small percentage change in Zp*,., 
then Zl remains essentially constant in magnitude 
and Rp becomes the principal variable. In any 
event, as Rp/Zj, decreases, the effective Q of Zl, as 
represented by an equivalent resistance and react- 
ance in series, must increase in order to compen- 
sate for the increased phase shift of Ip. 

1-273. From the qualitative discussion in the fore- 
going paragraphs it can be seen that in the con- 
ventional parallel-resonant crystal oscillators the 
state of an oscillator in operation is primarily de- 
termined by the impedance relations in the feed- 
back arm. Since the impedance of a crystal unit 
changes very rapidly with a small change in fre- 
quency, a crystal connected in the feed-back circuit 
makes the oscillator less critical in design than 
would otherwise be the case. Where maximum sta- 
bility is required, the vacuum tube will be operated 
as nearly class B as possible. Under class-A con- 
ditions, Rp and n are approximately given by the 
d-c, plate-characteristic data of the tube. In the 
case of power oscillators, amplifier operation will 
normally be class C, although class B or even class 
AB may be employed in particular circuits. In these 
cases, the effective tube parameters cannot be 
known beforehand, but reasonably accurate ap- 
proximations can be made and optimum operating 
conditions can be reached by more or less trial- 
and-error final adjustments. The operation of con- 
ventional oscillators is made less critical, both in 
starting oscillations and in maintaining a constant 
amplitude, by the use of gridleak rather than fixed 
bias. From the point of view of phase rotation, the 
conductance of the gridleak somewhat decreases 
the Q of Zb, and thereby necessitates increased 
detuning of the tank. Nevertheless, in low-power 
oscillations the gridleak losses can normally be 
considered negligible in comparison with the crys- 
tal losses. It is as a limiter and stabilizer of the 
amplitude that the gridleak bias is most important. 
Any changes in the circuit that tend to change E, 
automatically change the bias in such a direction 
that Rp and g„, of the tube are readily adjusted to 
new equilibrium values, so that the tendency is one 
of immediate opposition to the change in so far as 
the activity is concerned. 

TYPES OF CRYSTAL OSCILLATORS 

1-274. Crystal oscillators are frequently classified 

WADC TR 56-156 


Section I 
Crystal Oscillators 

as being either crystal-controlled or crystal-stabi- 
lized. A crystal-controlled oscillator is defined as 
an oscillator that cannot oscillate if the crystal is 
removed or is defective. A crystal-stabilized oscil- 
lator, on the other hand, operates as a “free-run- 
ning” oscillator if the crystal is removed. When 
the crystal unit is properly inserted and the “free- 
running” frequency is made to approach the nor- 
mal resonance of the crystal, the mechanical 
vibrations of the crystal sharply increase. At some 
point the piezoelectric effect will be sufficient to 
suddenly “capture” the oscillations and thereby 
synchronize them at the crystal-circuit frequency. 
Generally, the crystal-controlled oscillator is pre- 
ferred, since it is not desirable that oscillations 
continue if the crystal unit suddenly or gradually 
becomes defective. Also, crystal-controlled oscilla- 
tors are normally less critical to design and are 
less likely to jump suddenly from one frequency 
to another. The crystal-stabilized oscillator does 
have the possible advantage of being able to oper- 
ate successfully with very-high-Q crystal units 
whose piezoelectric coupling, however, would be 
too weak for the crystal to build up oscillations 
from a single initial impulse. A number of oscil- 
lator circuits appear to be border-line cases, that 
can only arbitrarily be classified as crystal-con- 
trolled or crystal-stabilized. For example, the Cl 
meter circuit shown in figure 1-106, if connected 
in the crystal position of Si, would fail to oscillate 
if the crystal terminals w'ere open, but not if they 
were shorted. Most of the oscillator circuits that 
are discussed in the following paragraphs are 
classified as crystal-controlled inasmuch as the 
oscillations do not occur if the crystal units are 
disconnected. 

1-275. A more practical classification from the 
standpoint of circuit design and of selection of a 
crystal unit is that of series- and parallel-resonant 
crystal oscillators. In general, the series-resonant 
type provides the greater frequency stability and 
can generate the higher frequencies; whereas the 
parallel-resonant type is the more economical to 
construct, can operate over a wider frequency 
range by the substitution of different crystal units, 
and can generate the greater power output. There 
are, nevertheless, a number of exceptions to the 
general rule. 

PARALLEL-RESONANT CRYSTAL 
OSCILLATORS 

1-276. The first quartz oscillators to find general 
usage as frequency-control devices were of the 
parallel-resonant type. These oscillators are used 
primarily with fundamental-mode crystals at fre- 


119 



Section i 

Crystal Oscillators 

quencies below 20 me. In the conventional circuits 
of this type, the crystal must operate between its 
resonant and antiresonant frequencies, thereby be- 
having as an inductor. Under these conditions the 
circuit does not oscillate if the crystal unit becomes 
defective. Unless a frequency monitor is to be 
available, these oscillators should be permitted a 
tolerance of 0.002 per cent or greater, depending 
primarily upon the tolerance rating of the crystal 
unit to be used. Under extreme operating condi- 
tions, an oscillator error of approximately twice 
the crystal unit tolerance should be permitted. The 
conventional circuits are of the Pierce and Miller 
types, or their modifications. Maximum stability 
is achieved with low crystal drive and class-A to 
class-B operation of the tube. Maximum power effi- 
ciency is achieved with class-C operation. Since 
fundamental-mode crystals become too thin and 
fragile for operation above 20 me, overtone crys- 
tals are necessary at these higher frequencies. 
However, the shunt capacitance, C„, and the series- 
arm L remain the same whether the thickness- 
shear crystal is operated at the fundamental or the 
overtone mode, whereas the series-arm C varies 
inversely with the square of the harmonic. Thus, 
the capacitance ratio, Co/C, increases with the 
square of the harmonic, so that the electromechan- 
ical coupling may be too weak to initiate oscilla- 
tions at normal voltages if the crystal unit is to 
be operated at parallel resonance. For this reason 
and also because Co, as well as the tube capaci- 
tances shunting the crystal, have larger suscep- 
tances at . the higher frequencies, which greatly 
reduce the operating range of the crystal unit, the 
parallel-mode circuits are unsuitable for use at the 
higher frequencies. If the basic circuits are modi- 
fied to employ series-mode crystals, or are used 
in conjunction with frequency-multiplying stages, 
they can provide stable control of very high fre- 
quencies. The introduction of a number of multi- 
plier stages with the attendant problems of pre- 
venting unwanted frequencies is usually less to be 
preferred than the direct generation of the end 
frequency by the use of overtone-mode crystals 
in series-resonant circuits. Although frequency 
multiplication involving more than one multiplier 
stage is still widely used in conjunction with par- 
allel-resonant master oscillators, this usage is 
found principally in medium- to high-frequency 
transmitters where various multiplying combina- 
tions can provide the maximum number of chan- 
nels with a minimum number of crystal units. 
Control by parallel-mode circuits of frequencies 
above 30 me is not very common. One example is 
to be found in Radio Set AN/ARC-IA. In this 


equipment, a fundamental frequency of 8 me, for 
instance, would be doubled in the oscillator plate 
circuit and increased nine times more in the plate 
circuit of the following stage. Thus, with only two 
tubes, a parallel-resonant oscillator can control a 
frequency of 144 me and higher. In the analysis 
of the particular oscillators to follow, the Pierce 
oscillator has been chosen as something of a refer- 
ence circuit as well as a point of departure in the 
discussion of many design considerations to be 
encountered in crystal oscillators. For this reason, 
the reader will find the treatment of the Pierce 
circuit, both qualitatively and mathematically, con- 
siderably more detailed than that of the other 
types of circuits. Space forbids as comprehensive 
a treatment for the other circuits, but the design 
problems and methods illustrated in the particular 
case of the Pierce oscillator are applicable in prin- 
ciple to all oscillatoi s. 

The Pierce Oscillator 

1-277. The Pierce oscillator is fundamentally a 
Colpitts oscillator in which the plate-to-grid tanl^ 
inductance has been replaced by a crystal unit, as 
shown in figure 1-120. The design of the Pierce 
oscillator is simpler and less critical than that of 
any other crystal circuit. As long as Ri and R* are 
large compared with the capacitive reactances 
shunting them, the Pierce circuit will oscillate with 
crystal units covering a wide band of frequencies. 
The use of load resistance, Ri, in figure 1-120 (A) 
aids in maintaining a reasonably flat response over 
a wide frequency range without the necessity of 
tuning adjustments other than the switching from 
one crystal unit to the next. Where a broad fre- 
quency range is not required, or where greater 
activity is necessary, an r-f choke should be used 
in place of Ri; otherwise, power approximately 
equal to I,,'‘Ri is simply wasted (Ib = average d-c 
plate current). But even with this economy, the 
Pierce oscillator cannot be used to generate as 
large an output as the Miller circuit. The principal 
reason is to be found in the fact that the imped- 
ance Z,,g of the equivalent circuit, which in this 
case is provided by the crystal unit in parallel with 
the plate-to-grid capacitance of the tube, must be 
approximately equal to, or greater than, Zp and Zg 
combined. Since the impedance of the crystal unit 
is fixed by its frequency and rated load capaci- 
tance, larger plate impedances are possible if the 
specified crystal impedance is Zg instead of Zpg. 
Thus, for the same crystal current, larger output 
voltages can be developed across the tank in the 
Miller than in the Pierce circuit. On the other 
hand, the effective feed-back phase Qf is greater 


WADC TR 56-156 


120 



S«cHon I 
Cryital Oscillators 


if the crystal is not connected between the grid 
and cathode. This permits the tank to appear more 
nearly resistive to the tube, so that fluctuations in 
Rp have less influence upon the frequency. Thus, 
a Pierce oscillator is generally more frequency 
stable than a Miller oscillator using the same crys- 
tal unit. The typical Pierce circuit employs a triode, 
although screen-grid tubes are usually to be pre- 
ferred, because the higher Rp and the negligible 
plate-to-grid capacitance serve to improve the fre- 
quency stability. The oscillator is generally used 
at frequencies above 200 kc and below 15,000 kc. 
If an overtone mode is to be excited the oscillator 
must be made frequency-selective, by replacing Ri 
with an inductor, Lj. The inductance of Lj must be 
such that the antiresonant frequency of Li in par- 
allel with Cl is lower than the operating frequency 
of the crystal, in order that Zp will appear capaci- 
tive. The use of the inductor-capacitor combination 
also reduces the harmonic content of the output 
waveform. If small L/C ratios are used, the effec- 
tive plate-to-cathode capacitance will be much 
greater for the overtones than for the fundamental 
frequency, so that the former are more readily 


bypassed to ground than would be the case if no 
coil were used. In deciding upon the type of oscil- 
lator circuit to use, those rule-of-thumb factors 
most favorable to the selection of a Pierce cir- 
cuit are : 

a. The frequency lies between 200 and 15,000 kc. 

b. The permitted frequency error is not less 
than 0.02 per cent, or 0.015 per cent if a regulated 
voltage supply is available. 

c. The oscillator is to be capable of untuned 
operation over a wide frequency range, simply by 
switching from one crystal unit to another. 

d. Only a small voltage output is required. 

e. The oscillator must be of inexpensive design. 

f. The oscillator must not be critical in opera- 
tion, but able to oscillate readily with relatively 
large deviations in the parameters of the external 
circuit. 

g. Wave shape is not critical. 

h. Same as above, except that the permitted 
frequency error is 0.01 per cent and thermostatic 
control of the crystal temperature is feasible. 

i. Same as above, except that the permitted fre- 
quency error is 0.005 per cent, thermostatic con- 
trol of the temperature is feasible, a regulated 



figure I-I20. Diagrams illustrating the equivalence (B) 

between the Pierce circuit and the Colpitts circuit colpitts oscillator 


WADC TR 56-156 


121 



Section I 

Crystal Oscillators 

ANALYSIS OF LOAD CAPACITANCE, C., 

IN PIERCE CIRCUIT 

1-278. Once that it has been decided to employ a 
Pierce type oscillator, the standard type of crystal 
unit is chosen which provides the desired fre- 
quency and frequency tolerance and which has 
been tested according to the Military Standards 
for parallel-resonance operation. One of the first 
design considerations is to ensure that the crystal 
unit will effectively operate into its rated load 
capacitance, C,. Such operation is necessary, else 
there can be no assurance that one crystal unit of 
the same type can replace another and still fall 
within the drive-level and effective-resistance spe- 
cifications. For most parallel-resonance crystal 
units the value of Ci is 32^/if, although at frequen- 
cies under 500 kc, values of 20 /i/if are common. 
In particular instances, still other values of Cx are 
designated. To a first approximation, referring to 
figure 1-121 (A), the crystal unit operates into a 
load capacitance equal to Cpg plus the total of Cg in 
series with the parallel combination of Ci, Cpc, and 
the effective inductive impedance presented by the 
vacuum tube. Since the Q of the feed-back arm is 
not infinite, Ep, it will be recalled, must be rotated 
slightly away from — /^Egi the direction is such 
that for a particular frequency Cpg -f Ci must be 
slightly larger than would otherwise be the case. 


Even though the actual equivalent tank circuit is 
slightly detuned, mathematically the crystal unit 
is to be in resonance with an effective load capaci- 
tance Cg. (See figure 1-108 (D).) The vacuum 
tube appears to the tank circuit as a negative re- 
sistance having a positive reactive component 
sufficient to cancel the excess susceptance of Zp. 
At equilibrium, the tube can be represented by an 
equivalent inductance, Lr, in parallel with & nega- 
tive resistance, pr, as in figure 1-121 (A). Note 
that pT is smaller than p of figures 1-121 (C) and 
(D). This is because pT is not connected directly 
across the crystal, but faces an impedance, ap- 
proximately Zl, that is less than the crystal PI. In 
figure 1-121 (B), Lt has been replaced by an 
equivalent negative capacitance, Cp. If Cpg can be 
considered negligible, X/ and R/ are equal to Xg 
and Rg, the equivalent parameters of the crystal 
unit alone ; otherwise, the values of Xg' and Rg' are 
based upon the assumption that the shunt capaci- 
tance, Co, of the crystal has been increased by an 
amount equal to Cpg. In figure 1-121 (C), Cp 
(= Cpg H- Cl) and C„ are shown combined into a 
single plate-to-cathode capacitance, Cp'. In figure 
1-121 (D) , Cp' and Cg are represented by a single 
load capacitance, C,'. If Cpg is negligible, C,' be- 
comes the equivalent load capacitance, C„ into 
which the crystal unit operates, and it should be 



i.m. PI.-. X.- ..j p...™ -k. 

!»»». .p.. -k. r if Ik. r .3....- »fc.. 

positive reactance equal to that of Lt- h « me r t piaio surr* 


WADC TR 56-156 


122 



equal to the value specified for the particular crys- 
tal. In any event (Ci' + Cp*) is the load capaci- 
tance that the crystal unit faces, and which should 
be equal to the rated value, C,. The value of C, 
includes not only the tuning capacitance in the 
plate circuit, but also the distributed capacitance 
of the output leads as well as the effective input 
capacitance of the next tube. C* is the input 
capacitance of the oscillator tube. The losses due 
to R„ the effective grid-to-cathode resistance, and 
to Rl. the effective load resistance, have been as- 
sumed to be negligible compared with those in the 
crystal unit. These assumptions can be made with- 
out appreciable error in a low-power Pierce cir- 
cuit that requires only a minimum of loading. 
From an inspection of figure 1-121 it can be seen 
that in order for a crystal unit to operate into its 
rated load capacitance, the design must be such 
that 

c. = Cp* -f C,' = Cp, -h 

1—278 (1) 

Effect of Cpg in Pierce Circuit 

1-279. Cpg effectively increases both the reactance, 
Xp, and the resistance, R„ of the crystal unit. This 
does not mean that the true effective Xe and Re are 
changed, for these are fixed by the fact that X* 
must resonate with the rated load capacitance, Cj. 
Still, insofar as the impedance from plate-to-grid 
is concerned, Xe and Re effectively have increased 
values which make it appear that the crystal shunt 
capacitance is increased by an amount equal to 
Cpg. This effect is not desirable, since Re is effec- 
tively increased by a greater percentage than Xp. 
(See’ equations (1) and (2) in figure 1-98 for the 
effect on Xe and Re if X, is held constant but Xco 
is decreased.) Thus, the larger the value of Cpg, 
the smaller the Q of the feed-back arm becomes 
the more the tank circuit must be detuned, the 
greater must be the value of the negative capaci- 
tance, C„, and hence the greater the frequency in- 
stability due to changes in the tube parameters. 
The plate-to-grid capacitance needs to be consid- 
ered only if a triode is used or if the crystal unit 
is oven-mounted. In the average triode Cpg is on 
the order of 1.5 to 2.5 fi/it, sufficient to increase 
the effective value of C„ by as much as 50 per cent 
in some cases. When the second grid of a tube is 
used as the oscillator anode, as in the case of penta- 
grid converters, Cpg is usually on the order of 1 n^tf. 
The pin-to-pin capacitance introduced by ovens 
may be as high as 5 /i/if. The Cpg of screen-grid 


Section I 
Crystal Oscillators 

tubes can all but be neglected, since the increase 
in capacitance across the crystal is only about one- 
thousandth of the total. Because of its negligible 
Cpg, a pentode is preferred when the frequency 
deviation must be kept to a minimum. In the re- 
maining discussion of the Pierce circuit, we shall 
assume that Cp* is negligible, so that X/ and Re' 
will represent the actual effective impedances of 
the crystal unit, and C/ will equal the rated capaci- 
tance, C,. Although we shall employ the unprimed 
symbols X, and Re to designate the plate-to-grid 
impedances, it should be remembered that this is 
only a convenience, for where X, is predetermined 
by the rated load capacitance and the frequency, 
Xe' necessarily increases or decreases, respec- 
tively, with increases and decreases in Cpg. Simi- 
larly, Cx' varies negatively with Cpg, but we shall 
assume that it is a constant equal to Cj. 

Determination of the Effective Negative 
Capacitance, C„, Introduced by Vacuum Tube 
in Pierce Circuit 

1-280. First, in order to avoid possible confusion, 
it should be pointed out that the selected reference 
or zero phase angle of the equivalent circuit in 
figure 1-119 is not the same as that implicitly as- 
sumed in the negative-resistance circuit of figure 
1-121 (A). In figure 1-119, the reference phase 
has been taken as the phase of Eg, whereas in fig- 
ure 1-121 (A) it is the phase of the current 
through the negative resistance pr (not Ip), which 
in turn is the same as the phase of the r-f plate 
voltage, Ep. Now, Ip, in the negative-resistance 
circuit, is physically the same as the Ip of the 
vacuum-tube generator circuit. The equivalent 
current through p-r represents that component of Ip 
in phase with Ep — not that part of Ip in phase 
with — /iEg. The current through the negative re- 
sistance is thus smaller in magnitude than the 
total r-f plate current. The imaginary current 
through Li-, or C„, is equal and opposite to that 
component of Ip which is 90 degrees out of phase 
with Ep. In the phasor chart in figure 1-119 (C), 
the bottom line shows the phase relations that are 
approached in a Pierce circuit if Rp of the tube 
approaches Zl in magnitude. Note particularly 
that Ep and Ip rotate in opposite directions. Ep 
must lag Ip by an angle whose tangent is at least 

as great as „ — ; that is, the tangent of the 

angle cannot be less than the reciprocal of the Qf 
of the feed-back arm. The minimum angle occurs 
when Rp is very much greater than the load imped- 
ance Zl and the gridleak losses are negligible. On 
the other hand, the phase difference between Ep 


WADC TR 56-156 


123 



section I 

Crystal Oscillators 

and Ip cannot be greater than 90 degrees, for the 
simple reason that Ep is a counter emf produced 
by Ip flowing through an equivalent impedance, 
Zl, which has no component of negative resistance. 
There is another limitation in that the rotation of 
Ip with respect to — /lEg cannot exceed 90 degrees 
minus the necessary rotation of Ep with respect to 
— /iEg. Otherwise, the necessary rotation of Ep 
cannot occur. As this extreme is approached, Zl 
approaches a pure reactance approximately equal 
to Xcp, Ip approaches a 90-degree phase lead over 
Ep, and the apparent Q of the entire plate circuit 


(approximately Zl/Rp) approaches the Q( of the 
feed-back circuit. These maximum and minimum 
phase angles are summarized in the following 
table. The phase angles are defined by the absolute 
values of their tangent expressed in terms of the 
phase Q of the feed-back arm, Q,. Also shown are 
the limiting Q’s of Zl and (Zl -|- Rp) : which re- 
spectively determine the phases of Ip with respect 
to Ep and — /lEg. In the last column are shown the 
limiting values of C„ which are discussed in the 
following paragraph. 


Angle of Ep at all times 

With Respect to : 

Q of; 

Cp 

(Approx) 

Ep 

— ftE, 

B 

Rp -|- Zl 

0® 

tan-‘(l/Q,) 



^■’(Of'+Rp + R.) 

Minimum angle of Ip 

t4n-‘(l/Q,) 

0” 

1/Qf 

0 

-Cp/Qf* 

Maximum angle of Ip 

O 

O 

tan-’(Qf) 

00 

Qf 

CpR, 

Rp + R« 


1-281. The approximate expressions given in the 
table above for the limiting values of C„ are de- 
rived upon the assumption that the unsigned phase 
angle between Ip and Ep is given by the equation 


tan' 


tan' 


-1 / Xzi, \ 

\ Rp Kzp / 


+ 


where Xzl and Rzl are the absolute values of the 
equivalent series reactance and resistance whose 


vector sum is equal to Zl; tan~> 


(i)" 


the un- 


tan-’ 


signed phase difference between Ep and — and 
’ ( p • ) is the unsigned phase difference 

\ Kp -f* iCzl / 

in the opposite direction between Ip and — 
From equation (1) it follows that 


Xz 


= tan tan 




1—281 (2) 


On applying the general trigonometric equation 
for the tangent of the sum of two angles 


tan (x -I- y) 


tan X ■+• tan y 
1 — tan X tan y 


Letting tan x = 1/Q( and tan y = 


Rp + Rj 


-, equa- 


1 281 (1) ^2) becomes 


Xzl 

Rzl 


1 Xzl 

Qf Rp -f- Rz 


1 - 


Qt (Rp + Rzl) 


Rp ~h Rzl ~h Qf Xzl 
Q( Rp -|- Qt Rzl — Xzi 


1—281 (3) 


On rearranging, equation (3) can be expressed as 
an equation for Rp ; 


Rp = 


Rzl^’ + Xzl" ^ Zl* 

Qf Xzl — “ Rzl Qf Xzl Rzl 

1—281 (4) 


WADC TR 56-156 


124 

























Figun 1-122. Gen«ralU«d Pierem osci/Zafor. Xz^ and 
Xcn npresant the potiiiva magnitudes of the equiva- 
lent reactances of the load and the dynamic effects 
of the tube, respectively, pr Zfie same as the pr in 
figure 1-121 


Note that for oscillations to be maintained, QjXz,, 
must be greater than Rzl, else Rp becomes nega- 
tive. Referring to figure 1-122, the effective ad- 
mittance of the Zl and Xcn branches in parallel is 


equal to - 


-. From this expression 


Rzl — jXzt jXco 
it can be shown by straightforward manipulation 
that if the reactive component of the admittance is 
to be zero 


Xc„ Xz, ^ Xz.* = 1-281 (5) 

On substitution of equation (5) in equation (4) 
and rearranging 


Xc. = Rp (q. - Ij) 1-281 (6) 

The values of C„ as listed at the end of paragraph 
1-280 are obtained from equation (6) by substi- 
tuting the values of Qf and Rzp/Xzp when these 
are expressed in terms of the basic circuit param- 
eters. In paragraph 1-289, it is shown that at 
equilibrium : 


Xcp + Xc. + X. -f = 0 1—289 (3) 

Rp 

The term, accounts for that part of the 

Kp 

negative capacitance which is necessary to com- 
pensate for the phase shift in Ip, but not for that 
part which compensates for the phase shift of Ep. 
For example, if Rp were infinite, the phase shift 

of Ip, would be zero, and likewise the term, 


Section I 
Cryctcil Oscillalory 

in equation 1 — 289 (3). Nevertheless, Ep must still 

be rotated by an angle equal to tan-'r^^, so that 

the tank cannot actually be parallel-resonant. 
Some value must be assigned to the negative ca- 
pacitance, for the apparent resonance to hold in 
the generalized negative-resistance circuit. Now, 


Letting 

and 


Zl 


Zp Zpip; 

Zp + Zp,c 


*Zp — jXcp 


Zpgo = Re + j(Xe -f- Xc*) 


we can express Zi, as a complex function equal to 
Rzt — jXzL, where Xzp is still assumed to be un- 
signed. Thus, 


(complex) Zl 



On multiplying both numerator and denominator 
by ^ Re 4- that 


Rzl = 


^Cp 


^Cp 


R. 




Re 


1—281 (7) 


and 


Xzp = 


Xcp I [ R, + (Rp + R.) ] 




Xcp I (Re Rp + Xcp^ ) 


Re Rp 


1—281 (8) 


Xcp I Rp’* 


So 

Rzl _ 

X*L " R,Rp* + Xcp* (Rp + R.) 
I Xcp I Rp 


R R 4- X ^ ^-281 (9) 

•Ke-Kp Acp 

Also, when assuming that the grid losses are negligible, 

X. 4- Xc* ^ I Xcp I (Rp 4- Re) 

R, Re Rp 

1—281 (10) 


Q. = 


WADC TR 56-156 


125 




section i 

Crystal OKillatart 

The expression on the right in equation (10) is 
obtained by substitution from equation 1 — 289(3). 
On substituting equations (9) and (10) in equa- 
tion (6), we have 


Xc„ = 

I Xf I r ^P ~l~ ^P* 1 

' ' L Re R. Rp -!- Xcp* (1 + Re/Rp)J 

Xc„ = I Xcp I 

[ RpXcp" (l+Re/Rp)-hReX-t>R.Xcp^ (l-hR./Rp) ~| 
L Re' Rp + ReXcp' (1 + Re/Rp) J 

1—281 (11) 

Equation (11) is obtained by using the exact 
values of Rzl/Xzp and Qj as given by equations (9) 
and (10). Although the equation for Xcn involves 
the difference between two nearly equal terms, the 
error introduced by using the approximate values 
of Qf and Rzi,/Xzl is negligible for all practical 
purposes as long as Rp >> R,. 

Now, 


Cp = 


1 

wXcn 


SO, on substitution of equation (11), 


Cp — “ Cp 

r Re' Rp -h R. Xcp' (1 + Re/Rp) 1 

LRpXcp' (l+Rp/Rp) -I- R.'Rp + RpXcp* (H-R./Rp)J 

1—281 (12) 


In the practical case, Rp is much greater than R, 
and X cp’ is much greater than R,’, so that the ap- 
proximate equation for Cp becomes 


r - r r R.’’ Rp + R. Xcp' 1 

~ Rp (Xcp' -h R.') J 

The last term on the right inside the parentheses 

Xrn* 

is obtained by assuming that ^ < 7 p « is more 

Aop “H XV* 

It 

nearly equal to than to 1. We make this 

iVp -f- K* 

assumption arbitrarily for the convenience in re- 
membering the limiting values of Cp. That part 
of Cp which is necessary so as to compensate for 
the phase shift of Ep with respect to — /lE, is ap- 
proximately equal to — Cp/Qf*; whereas the part 
necessary to compensate for the phase shift of Ip 


with respect to — is approximately equal to 
^ 

s — r^rr- R is this latter component that is ac- 

Kp + K* 

X R 

counted for by the term, * , in equation 1 — ^289 

Kp 

(3). By equation (12), when Rp/Zl approaches 
infinity, the limiting value of C„ is found to be 


Cp 

(Rp/Zl -» oo) 


-CpR,' 
Xcp' + R.' 


^ 1-281(14) 

Wi 


When Rp/Zl approaches its smallest possible 
value, the ratio Rzp/Xzp becomes negligible com- 
pared with Qt, so that Xcb by equation (6) is ap- 
proximately equal to QfRp. Substituting equation 

(10) for Q„ we have, when / — 

\ Rp -f- Rzl / 

“ «Q,Rp 

R. 

w| Xcp|(Rp -l- R.) 


— CpR, 
Rp + R« 


1—281 (16) 


As indicated in figure 1-121, the total effective 
plate capacitance is 

1—281 (16) 


When Rp > > Zl, the Q of Zl approaches 1/Qt in 
value and 


Cp'=Cp(l--^) 1-281(17) 

When Rp is small compared with XcpVR*. then 
Rzl/Xzl becomes small compared with Qf and 

Cp' = 1—281 (18) 

Rp + R, 

Rp is normally much greater than R,. Only low- 
frequency crystals have effective resistance 
which approach in value the plate resistance of 
low-power vacuum tubes. For all practical pur- 
poses in the average Pierce circuit, Cp' can be as- 
sumed to equal Cp, the static plate capacitance, 
except when considering problems of frequency 
stability. What is important to note in the limiting 
equations for C, is the fact that if the tank is to 


WADC TR 56-156 


126 



Saction I 
Crystal Oscillators 


be operated well off resonance, Rp becomes quite 
an important factor in determining the frequency. 
In this case, because C„ is relatively large, any 
variation in the tube R„ has a great effect upon 
the frequency. It should be remembered that the 
parameter Qt has been used to account for the re- 
quired rotation of Ep with respect to —/j-Eg. Inso- 
far as the gridleak is effective in increasing the 

necessary phase shift, Qt = — ^ cannot be 

assumed, and the complete equation 1 — ^272 (1) 
must be used. 

THE EFFECT OF R, UPON THE 
FREQUENCY OF PIERCE CIRCUIT 
1-282. It can be seen that for large values of the 
Ri,/Zi, ratio, C„ is small and its percentage varia- 
tion with changes in Rp is smaller still. Cp is, effec- 
tively, a frequency-determining parameter, but 
more exactly it is a mathematical function that in- 
directly expresses the effect of Rp and Qf upon the 
frequency. The smaller the Qt and Rp, the larger 
is C„; and the larger C„, the greater is the effect 
of Rp. Since Rp is subject to change with changes 
in the tube voltages, tube aging, and the like, it is 
important to keep C„ as small as possible. This can 
be done by designing the circuit to operate with 
as high of value of tube Rp as is practicable. For a 
given tube, the higher values of an effective Rp are 
to be obtained when the tube is conducting during 
only a small fraction of a cycle. This in turn re- 
quires that the oscillator tube be operated class C, 
so that a larger grid bias than otherwise is re- 
quired. However, if the crystal drive level is to be 
kept low and if the gridleak is to have a negligible 
effect on the effective Q(, and hence upon C„, the 
gridleak resistance must be as large as practicable 
without running the risk that the tube will block 
or operate intermittently. The limiting value of Rp 
occurs when — is just sufficient to maintain 
oscillations. If the vacuum tube could operate into 
a pure resistance. Ip would be in phase with — /xE*, 
and Rp would be eliminated as a frequency-deter- 
mining element. In the conventional Pierce circuit 
this could occur only if Qt were infinite. 
Pkase-Stabilized Pierce Circuit 
1-283. If an inductor is inserted in the plate cir- 
cuit of the oscillator, as indicated in figure 1-123, 
having a reactance equal and opposite to the effec- 
tive reactance Xz,., then Ip undergoes no phase 
rotation, and changes in Rp, although affecting the 
activity, will have little effect upon the frequency. 
With Ip in phase with — /xEg, the Q of Zl must equal 
1/Q,, and the operation of the tank is the same as 
it would be if Rp were infinite. If Qf is reasonably 
large and is approximately equal to IXco/R*!, 

R*x = Zi, = XcpVRe = ! Xcp IQ. 


Ip iXu-iXZL 



Figure I-I23. Coil inserted in plate circuit of Pierce 
oscillator to prevent phase of ip front being influenced 
by changes in Rp 

Since 

Xzp/ Rzi. = 1/Qf 

XzL is approximately equal in magnitude to Xcp. 
Thus, for the vacuum tube to look into a resistive 
load, the inductor should have a reactance approxi- 
mately equal in magnitude to X,-,,. This value as- 
sumes that the gridleak and output losses are 
negligible. When such assumptions cannot be 
made, the value of the series plate reactance be- 
comes a more involved function. Llewellyn ana- 
lyzed this type of circuit and eliminated Rp from 
the frequency-determining equation (phase rota- 
tion equation) by equating the sum of the factors 
of Rp to zero. Although the approach is different 
and the grid losses are assumed to be predominant, 
Llewellyn’s mathematical elimination of the ef- 
fects of Rp upon the frequency by the introduction 
of a plate inductor in series with the tank appears 
to be equivalent to the qualitative condition that I„ 
must be held in phase with — juE,,. The experi- 
menter, nevertheless, should be warned that the 
theory of this type of stabilization has been ana- 
lyzed above, and also by Llewellyn, only in terms 
of the phase relations. Difficulty will probably be 
experienced in obtaining .stable oscillations with- 
out additional modifications to ensure that the 
limiting • characteristics are changed from a 
voltage- to a current-controlled nature. This fea- 
ture of oscillator theory has not been fully ex- 
plored, but see paragraphs 1-585 to 1-598 for a 
general discussion, and paragraph 1-323 for a 
particular example of an attempt, which was not 
entirely successfu', to . phase-stabilize a Pierce 
circuit. 

Conditions for Maximum Rp in Pierce Circuit 
1-284. Referring to figures 1-119 (A) and (B), we 
shall begin with the assumption that the tank is 

Z ^ 

operating near resonance so that Zl«-^— , where 

R, (not shown) is the effective resistance of the 
crystal unit whose total impedance is represented 
by Zl> therefore, is very nearly resistive, and 
Ir is approximately equal in magnitude to the cur- 
rent through Zp. 

Thus, Ig « Ep/Zp 


WADC TR 56-156 


127 



Section i 

Crystal Oscillators 


also.E, = I.Z. = 

Rearranging, Ep/E, ^ = C,/Cp 

In the interest of maxium stability it is desirable 
for Rp to be a maximum. The problem is to find 
what capacitance ratio, Cg/C,„ permits the largest 
possible value of Rp consistent with the rated drive 
level and load capacitance of the crystal unit. The 
phase-rotation equations do not enter the problem 
— only those equations that concern the magnitude 
of the equilibrium voltages and currents are of 
concern now. The crystal specifications indirectly 
set an upper limit for the tank current, I,. Thus, 
the output voltage, Ep I* Zp, also has an upper 
limit, since Zp (= Zp* — Zg) has a theoretical maxi- 


AndL = 


_ mE, 


pEpZg 


Rp + Zl Zp(Rp -h Zl) 


SoEp = — 

^ Zp(Rp -f- Zl) 


or fiZgZi, — RpZp — ZpZj. 


On substituting (Rpgm) for n, where gn is the 
transconductance of the tube. 


RpgniZgZl, — RpZp — ZpZi. 

orRp = ^ 

gmZgZi, — Zp 


1—284 (1) 


Dividing both sides by Zl, we have 


mum equal to fiZg = which is ap- 

proached as Rp approaches zero. At the other ideal 
extreme, Zp and approach zero and the Rp/Zi, 
ratio becomes very large. Now, a large Rp/Zt is 
desired, but some compromise must be made, since 
the Q of the feed-back circuit becomes increasingly 
small as Zg approaches Zpg in magnitude. A rigor- 
ous treatment of the problem to find that relation 
between Rp/Zl and Qf that provides an optimum 
frequency stability would require that a complete 
equation of frequency stability be established and 
that those impedance relations be determined that 
produce a minimum frequency deviation for small 
changes in the circuit parameters. Equation (2) in 
paragraph 1-288, which is a first order expression 
for the fractional change in frequency for a change 
in Rp, indicates that the percentage deviation in- 
creases directly with the first power of Cp, and 
inversely with the second power of Rp. This sug- 
gests that the stability increases as long as Cp/Rp* 
decreases with an increase in Cp, and is a maxi- 
mum at the value of Cp, if existent, at which this 
ratio begins to increase. Such an approach will not 
be attempted here. Unless all the characteristic 
curves of a vacuum tube are available, so that 
either /x, Rp, or gn, can be used as an independent 
variable to eliminate the other two from the equa- 
tions, concrete conclusions cannot be reached con- 
cerning the optimum design of an oscillator using 
that particular tube. A more qualitative analysis 
is presented below, and although the indicated 
optimum relations cannot be considered conclusive, 
they can serve as first approximations. All imped- 
ance, current, and voltage symbols given below are 
considered positive and undirected. 

Now, Ep = Ip Zl 


Rp Zp ^ 1 

Zl gmZgZL — Zp gmZgZL _ 

Our present concern is to seek the largest practical 
value of Rp/Zl, so that the phase of Ip will be least 
affected by small changes in Rp. Now, Z* «« Zp, — Zp, 
where Z^ represents the predetermined crystal 
impedance, which is approximately equal to X,. 
Also, Zl «= Zp*/Re. On substitution of these values 
in the equation for Rp/Zl, it is found that 

Rp R. ^ R; 

Zl gmZpZg — Re g,„(Zp(Zp — Zp*) — R* 

1—284 (2) 

It can be seen from equation (2) that for oscilla- 
tions to be maintained gm (ZpgZp — Zp’) must be 
greater than R,. A maximum Rp/Zl ratio is ' 
approached as the product gm(Zpg Zp — Zp’) ap- 
proaches the value of R*. Of course, it is impossible 
for the denominator in equation (2) to be actually 
equal to zero, for then Rp would be infinite; but it 
is plausible to assume that a denominator much 
smaller than the value of R. can be realized. Thus, 
we can write 


(optimum) g„ = 


R. 

ZpgZp - Zp* 


Re 

ZpZg 

1—284 (3) 


The more nearly this equality is approached, the 
greater will be the frequency stability. The ques- 
tion arises, is it preferable to seek this equality 
with a small or a large value of g„ ? Assuming that 


WADC TR 56-156 


12 « 



£ 

(Rp/Zl) >> 1, the equation, Ip = , can be 

written approximately Ip = f»Eg/Rp = gmE,, or 
gm = Ip/Eg. A large g„ means a large r-f plate 
current for a given excitation voltage. This would 
be desirable from the point of view of maximum 
output, but an examination of the denominator in 
equation (3) shows that a large transconductance 
means that the plate impedance, Zp, or the grid 
impedance, Zg, must be made small if the equation 
is to hold. A small Zg (large Z„) means a large 
Zl and also a large E„/Eg ratio. Both consequences 
are incompatible with a large Rj/Zl ratio. The 
former is obviously so, and the latter is implicitly 

so because the ratio of Ep/E, times ^ 


k = must equal /i. Every increase in Ep/Eg 

erefore requires an approximately proportional 
decrease in the Rp/Zl ratio, insofar as n can be 
assumed to remain constant. On the other hand, 
a large Zg and small Z, permits a large Rp/Zl ratio 
and has the additional advantage of permitting a 
given excitation voltage with a minimum crystal 
current. It is under these conditions that equation 
(3) will be most nearly exact. There are serious 
disadvantages, however, when operating at a max- 
imum Rp/Zl ratio; the most important of which 
is that the Q of the feed-back circuit rapidly de- 


creases as Zg is increased, since Q( < 


Zpg — Zg 

R. 


Also, 


the voltage output is weak, and has a tendency to 
instability. This will be discussed more fully later. 
Since the excitation voltage is stronger for a given 
crystal current, the grid losses increase propor- 
tionately and may no longer be negligible. Fur- 
thermore, unless the tube is operated class C, the 
power efficiency is very low. These last mentioned 
disadvantages, nevertheless, are minor compared 
with the effect on Qf. The minimum effective Q of 
the average crystal unit when operating into its 
rated load capacitance is not unduly large. A grid- 
to-cathode reactance equal in magnitude to three- 
fourths Xe reduces Q, to one-fourth Q«. Since the 
purpose of a large Rp/Zt ratio is to permit the 
entire plate-circuit impedance (Rp -f- Zl) to ap- 
pear as nearly resistive as possible, the better 
stability risk is to operate the parallel-resonant 
oscillator with small rather than large Ip and gp,. 
Since we are assuming that >« Ip/Eg, it can be 
seen that the smaller the value of gm, the smaller 
is the r-f plate current for a given excitation volt- 
age, or, for a given plate current, the smaller the 
value of g«, the greater the excitation voltage. The 
problem becomes one of determining what capaci- 
tance ratio, Cg/Cp, permits the smallest possible 


Section I 
Crystal Oscidatars 

gm. By equation (3), gm is a minimum when the 
denominator of the right-hand term is a maxi- 
mum. Since the impedance of the crystal unit, Zp„ 
is to be held constant, (Zp -f- Zg) is also a constant. 
Thus, the product Z,Zg can easily be shown to be 
a maximum when 


Zp = Zg = Zpg/2 1—284 (4) 

A maximum operating Rp and a minimum Ip with 
a given excitation voltage can thus be obtained 
when the capacitance and voltage ratios are 

Ep/Eg = Cg/Cp = 1 

It is quite fortunate that gm has a minimum value. 
At all other operating values a small change in 

Q 

the jr ratio causes gm and Rp, and hence the fre- 
quency, to change. At the minimum gm the rate of 
change in the tube parameters is necessarily zero, 
so that the stability in this respect is a maximum. 
When the more exact equation 1 — 289 (2) is used 
instead of equation (1) above, and when /t/Rp is 
substituted for gm, it can be shown that 


•D _ Xcp Xe -1- Xcp Xcg 4- /iXcp Xcg 

^ r: 

1—284 (5) 

Note that Rp, as long as is constant, is inversely 
proportional to R,. Now, approximately 


Xcp = - (X. -h Xcg) 

Substituting in equation (5), Rp becomes a func- 
tion of X,, Re, n, and Xc*. Assuming that the first 
three parameters are constant, it can be shown 
that Rp is a maximum when 



X. (m -h 2) 
2 (/i -H 1) 



1 + 


P+ 1/ 

1—284 (6) 


If (/* + 1) >> 1, equation (6) states approxi- 
mately the same conditions as does equation (4) . 
If /I is small, equation (6) should be accepted as 
the more accurate in computing the optimum 
Cg/Cp ratio, since a minimum gm coincides with a 
maximum Rp if the d-c plate voltage is kept con- 
stant. The capacitance ratio and values corres- 
ponding to equation (6) are 


Cg/Cp 


ft 

M 2 


1—284 (7) 


WAOC TR 56-156 


129 



Section I 

Crystal Oscillators 
or 


C. 


and 


Cp 


2C, (m + 1) 
(m + 2) 


2C, (m + 1) 


1—284 (8) 


1—284 (9) 


Under these conditions the excitation voltage be- 
comes greater than the voltage across the plate 

load by a factor of — , and the following addi- 
tional relations hold: 


(max) Rp = (m + D Zl 1—284 (10) 

(min) g„, = ^ 1—284 (11) 

tVp 

As a practical consideration in design as well as 
for the sake of simplicity in discussion it is con- 
venient to assume that the optimum Cg/Cp ratio is 

equal to one rather than — However, in inter- 

/I j 

preting the equations above, a word of caution is 
necessary. Returning to equation (2), it will be 
seen that the maximum to be sought for Rp/Zi, is a 
“practical,” not a “mathematical” maximum in the 
sense that a curve of R„/Zi, rises to a peak and then 
decreases. The curve of equation (2) plotted against 
Zp passes from positive to negative infinity as the 
denominator passes through zero and thus is dis- 
continuous at that point. However, for any given 
value of gm sufficiently large for Rp/Zt to be posi- 
tive, the curve does have a true minimum, not a 
maximum, at the point where ZpZg is a maximum. 
To avoid confusion as a result of this apparent 
contradiction, it is important to recall that the 
“practical” maximum is to be sought by making 
equation (3) as nearly true as possible, and not 
by the process of making Zp = Zg. This latter con- 
sideration is in the interest of over-all stability 
and maximum activity (if measured by the d-c 
grid current) for a given d-c plate voltage and 
load capacitance. Another point that should be well 
understood is that the minimum g„„ minimum n, 
and maximum Rp, are all coincidental. From the 
point of view of frequency stability the real in- 
terest is in the maximum Rp. From equation (10) 
it can be seen that the magnitude of the maximum 
Rp will increase with but remember that this 
value of /i is the minimum obtainable with a given 
tube and Ep. As will be discussed more thoroughly 
in paragraphs 1-294 and 1-295, an oscillator vac- 
uum tube cannot be operated so that is the maxi- 


mum possible without the risk of amplitude 
instability. Thus, class- A operation where the tube 
is operated only along the straight portion of the 
E,Ib curve is not feasible in gridleak oscillators. 
Understand that if equation (7) holds, equations 
(10) and (11) automatically hold. in each equa- 
tion is the effective n when equilibrium is reached 
and is not the starting It is the minimum n that 
can be obtained as long as the crystal resistance. 
Re, and the total load capacitance, Cg, remain con- 
stant. Because Rp, gm, and /i all pass through ex- 
tremes at the optimum capacitance ratio, it might 
be thought that the operating conditions are more 
ideally unique than they actually are, because the 
instantaneous rate of change for all the tube 
parameters with the capacitance ratio is zero 
under these conditions. Remember, however, that 
these maximum and minimum values apply only 
in the event that the total C, remains constant. An 
independent variation in Cg or Cp will cause the 
frequency to change, and the tube parameters will 
vary. For instance, gm will tend to vary directly 
with both Cg or Cp. Only when Cg and Cp are 
adjusted simultaneously so as always to maintain 
the same total load capacitance will the instanta- 
neous changes in the tube parameters be zero as 
the capacitance ratio is varied through its opti- 
mum value. 

1-285. If equation 1 — 284 (3) is expressed as a 
function of Zp*, by substituting from equation 
1 — 284 (4), the minimum value of g^ becomes, 
approximately, 

4R 

(min) g™ = ^ 1—285 (1) 

Since Zpg is the crystal impedance, approximately 
equal to Xp, the minimum value of gn, can be ex- 
pressed as 


(min) g„. = ^ 1—285 (2) 

where PI is the performance index. Using a one- 
to-one capacitance ratio and a vacuum tube of high 
Rp, equilibrium will be reached at the value of gm 
defined in equation (2). Such operation generally 
provides the maximum frequency stability in a 
Pierce oscillator. In estimating the value of PI, 
Xp is numerically equal to the reactance of the 
rated load capacitance, and Rp must be assumed 
to be the maximum permissible effective resistance 
according to the military specifications of the 
crystal unit being used. 


WADC TR 56-156 


130 



CAPACITANCE RATIO, C,/C„ FOR GREATER 
OUTPUT IN PIERCE CIRCUIT 

1-286. Where a maximum output consistent with 
the minimum frequency-stability requirements is 
desired in a Pierce oscillator, the Cg/Cp ratio can 
be increased and a vacuum tube providing a large 
transconductance and a large amplification factor 
should be used. The first consideration is that Ig 
must not exceed a value that would cause the 
power dissipation in the crystal unit to exceed the 
specified drive level. If the output of the oscillator 
is capacitively coupled to the grid of a buffer ampli- 
fier, the output power becomes a minor consider- 
ation compared with the output voltage. If this 
voltage is to be a maximum for a given tank cur- 
rent, the plate impedance Z,, must be a maximum. 
This means that the capacitance ratio, Cg/Cp, must 
be as large as practicable. The larger this ratio, 
however, the smaller will be the excitation voltage 
for a given Ig. The smaller the excitation voltage, 
the smaller will be the gridleak bias, and conse- 
quently Rp will be less whereas gn, and will be 
greater. An examination of equation 1 — 284 (3) 
reveals that the required magnitude of gm becomes 
very large as Zp approaches the value of Zpg of the 
crystal unit. Where a relatively large frequency 
deviation can be tolerated a large Rp may not be 
necessary, so that increased voltage outputs can be 
obtained with tubes of high transconductance at 
low d-c plate voltages. In any event the Cg/Cp 
ratio can never exceed the amplification factor of 
the tube, nor can the r-f plate voltage be greater 
than, nor equal to, the voltage across the crystal 
unit. It should be understood that the higher volt- 
age outputs are only to be had with a large Cg/Cp 
ratio because of the limitations on the crystal drive 
level and the load capacitance, Cg, and are not due 
to the fact that the Pierce or Colpitts type of cir- 
cuit is inherently more active when the Cg/Cp ratio 
is a maximum. To the contrary, with a fixed 
C., maximum amplitude of oscillations is to be 
obtained when Cp = Cg = 2C,. Where a larger 
than minimum power rather than voltage output 
is required, this can probably best be achieved 
when Cg/Cp lies between 1 and 2, and in this case 
a larger g„, is necessary to maintain oscillations. 
When a relatively large power output is required, 
the Pierce circuit should not be used. 

HOW TO ESTIMATE THE FREQUENCY 
VARIATION AND STABILITY OF A 
PIERCE OSCILLATOR 

1-287. In paragraph 1-243 it was shown that the 
frequency-stability coefficient, Fx., of the crystal 
unit is defined as the percentage change in X, per 


Saction I 
Crystal Oscillators 

percentage change in frequency. By equation 

Op t 

1 — 243 (1), Fxe = ^ ^ . The reciprocal, 1/Fx„ is 

thus equal to the percentage change in frequency 
per percentage change in reactance. Since X, is 
equal numerically to the reactance of the total load 
capacitance C„ the fractional change in the load 
reactance multiplied by 1/Fx,. will give the frac- 
tional change in frequency. Thus, 


^ ^ _ J_ ^ _ J_ 

f “ w " Fx. ■ Xg " Fxe ■ Cg 

1—287 (1) 


In the Pierce circuit, if it can be assumed that the 
interelectrode plate-to-grid capacitance and the 
grid and output losses are negligible, Cg will equal 


r> } , r where Cp' = Cp Cp. (See figure 1-121 

'-'p + t/g 

(C).) If it is desired to find the fractional change 
in frequency for a small change in, say Cp', 
equation ( 1 ) can be used by expressing dCg as a 

function of dCp', thus : dCg = dCp' • 

Qv/n 


With 

_ Cp'Cg, 

- “ Cp' + Cg 

then 

dCg ^ Cg^ 

dCp' (Cp' -H Cg)^ 

and 

dCg Cg^dCp' / Cp'Cg 

Cg " (Cp' -t- Cg)^ / Cp' -h Cg 

^ Cg dCp' ^ _c^ , 

Cp' (Cp' + Cg) (Cp')^ 


On substituting in equation (1) : 


f 


1 

Fxe 


c, 

(Cp')^ 


dC„ 


Likewise, 



In the event that Cp' =r C, = 2Cg 
Equations (2) and (3) become 


df dCp' 
f “ 4 Fxe 


1—287 (2) 


1—287 (3) 


1—287 (4) 


WADC TR 56-156 


131 



Section I 

Crystal Oscillators 


^ = - dCr 

f " 4Fxe 


1—287 (5) 


1-288. When a more detailed expression of the fre- 
quency deviation is desired C,,' can be replaced by 
(C„ -|- C„), and Cn, in turn, can be expressed as a 
function of its variables. A rigorous analysis of 
the effect of each parameter upon the frequency 
would be quite involved. Probably the simplest ap- 
proach for determining the frequency deviation 
due to a change in some particular circuit param- 
eter would be to begin with an appropriate 
equation in paragraph 1-287, and express the 
differential element as a function of the differential 
of the particular circuit parameter. This is the 
method that was used when dC, in equation 1 — 287 
( 1 ) was expressed as a function of dC/ and of dC* 
in equations 1 — 287 (2) and 1 — 287 (3) , respective- 
ly. As an additional example, suppose4;hat it is de- 
sired to determine approximately the frequency 
deviation due to a change in Rp of the vacuum tube. 
Let it be assumed that C* = 2Ci. The most appro- 
priate equation to begin with is 1 — 287 (4), since 
dC„' can be expressed as a function of dRp. The 
problem is to determine the function that gives 
the change in C,,' due to an infinitesimally small 
change in Rp, and to substitute that function for 


dC„ 


dCp' in equation 1 — 287 (4) . Since dCp' =:-gj^'dRp, 

the first step is to determine dCp'/dRp, and then 
simply to multiply this by dRp. Now, by equation 


1—281 (16), 


Cp' = Cp - Cp/Q,^ 


Cp Rp 

Rp + Re 


so 

dCp'/dRp = CpRe/(Rp + Re)' 


or 

dCp' = — • dRp 1-288 (1) 

Substituting this function for dCp' in equation 
1—287 (4) ; 


df Cp R, dRp 

f 4(Rp -H RJ' ■ Fxe 


1—288 (2) 


Note than an increase in R„ causes a decrease in 
the frequency. It should be well understood that 
the equations above are only rough approxima- 
tions. For example, one of the approximations in 
equation (2) is the assumption that Rp and R, 
are independent variables, which, of course, is not 
true. However, the direct effect of a change in Rp 


upon the frequency can be considered much greater 
than the indirect effect due to a change in R, re- 
sulting from the initial change in frequency. The 
differential element, dRp, in equation (2) can, in 
turn, be expressed as a function of an infinitesimal 
change, dEp, in the d-c plate voltage. The general 

equation for this function is dRp = 4^ • dEp. How- 

ever, the derivative term, dRp/dEp, will be quite 
difficult to determine, except by experiment, since 
it depends upon both the tube and circuit char- 
acteristics at the operating voltages. For its 
mathematical expression the principal consider- 
ations would be the change in Rp with a change 
in Ep, assuming a constant grid bias, the change in 
grid bias due to the change in plate voltage, and 
the change in Rp due to a change in grid bias, as- 
suming a constant plate voltage. In using the 
equations above it is only necessary to substitute 
small finite changes in the independent variable 
for its differential. For example, if the input ca- 
pacitance were to decrease an amount aC, = 
the fractional change in frequency would be given 
approximately by equation 1 — 287 (3) if we were 
to substitute — for dCg. 

ENERGY AND FREQUENCY EQUATIONS OF 
PIERCE CIRCUIT AS COMPLEX FUNCTIONS 
OF LINEAR PARAMETERS 

1-289. It is beyond the scope of this handbook to 
present the more rigorous analyses of the various 
oscillator circuits. These can be obtained from the 
various reference sources listed in the index. 
Actually, even when following the more explicit 
equations, so many approximations must be made 
for the sake of simplicity, and so many unknowns 
are involved, such as stray circuit capacitance, 
that the final solutions can rarely be considered 
more than general indices of the actual circuit con- 
ditions. If maximum mathematical exactitude is 
desired in determining the frequency and activity 
characteristics of an oscillator, the analysis should 
be performed by differential equations assuming 
nonlinear parameters. Such equations are quite 
involved and are rather difficult to interpret qual- 
itatively. If a moderately rigorous analysis is de- 
sired, the equivalent circuit in figure 1-119 (B) 
may be assumed to have linear parameters, as has 
been assumed in our previous discussion, but in- 
stead of handling the impedances as real numbers, 
to represent them by complex functions. For the 
Pierce circuit 

Zp = jXcp 

Zg = jXcg 

Zpg= R. + jX, 


WADC TR 56-156 


132 



An equation expressingr the conditions for oscilla- 
tion is derived very similarly to the method fol- 
lowed in paragraph 1-284, except that I, is 
expressed by the exact equation 


Section I 
Crystal Osdllators 

necessary for the feed-back energy to be sufficient 
and stable — in other words, for the loop gain to 
equal unity. The equation for the imaginary part 
when made equal to zero can be expressed as: 


I = ” 

‘ Zp. -h Z, 

rather than by the simplifying equation 

I. = Ep/Zp 

For either the Pierce or Miller circuit, the condi- 
tions for stable oscillation are expressed by the 
equation 


Xcp + Xc, + X. -h = 0 1-289 (3) 

When this equation holds, the loop phase shift is 
zero. Whereas equation (2) is said to define the 
feed-back energy requirements, equation (3) is 
said to define the frequency requirements. Note 

that in equation (3), the term is equivalent 

to the reactance of a dynamic capacitance 


-Rp 


Zp (Zp, -b Z.) 
gm Zp Zj -f- Z, 


1—289 (1) 


/-I Rp 


1—289 (4) 


where Z, = Zp -|- Zpg -f Z^ Elquation (1) is similar 
to equation 1 — 284 (1) except that in the derivation 
of equation (1) I, is expressed by its exact func- 
tion and the impedances represent complex quan- 
tities. When the impedances are expressed as 
complex functions, the right-hand side of equation 
(1) can be reduced to the sum of a real quantity 
and an imaginary quantity, each with the dimen- 
sions of impedance. 

Thus, Rp = R 4- jX 

However, since Rp is not, itself, reactive, the 
imaginary term, jX, must equal zero, and the real 
term, R, must equal Rp. In this manner two equa- 
tions, X = O and R = Rp, involving the same 
variables are obtained, both of which must hold 
if stable oscillations are to be maintained. A mini- 
mum of two equations is necessary, since there are 
two independent functions to perform. One func- 
tion is to fix the frequency so that the excitation 
voltage is properly phased, and the other function 
is to ensure that the feed-back energy per cycle is 
exactly equal to its dissipation per cycle. If both 
sides of equation (1) are divided by Rp, the right- 
hand side again reduces to the sum of real and 
imaginary parts. The real part must be equal to 
1, and the imaginary part must again be zero. It 
can be shown that the real part becomes, after 
multiplying through by Rp : 


which can be imagined to be in series with Cp. The 
capacitance of the combination becomes 


Cp Cd _ Cp Rp 

Cp + Cd Rp -l* Re 


1—289 (5) 


which is exactly the same as the small-Rp value 
of Cp' in equation 1 — 281 ( 18) . Ca is thus a positive 
series dynamic capacitance equivalent to part of 
the negative parallel dynamic capacitance Cp. 
Equation (3) indicates that as Rp increases in- 
definitely Xrp -|- X( g + Xe -♦ 0. This is not to be 
interpreted as meaning that the tank circuit ap- 
proaches a parallel-resonant state as a limit or 
that the total dynamic capacitance approaches 
zero. Actually, even if the sum of the first three 
reactances did equal zero, the tank would not be 
at resonance because of the presence of R, in the 
feed-back arm, and a dynamic capacitance would 
need to be effectively present. What equation (3) 
does show is that, as Rp increases indefinitely, the 
frequency becomes entirely determined by the 
tank-circuit parameters. In the limit, Xc* X, = 
— Xcp. As this state is approached, the impedance 
of the feed-back arm can be represented as 


Zpgc R« “1“ jXpgc Re jXcp 

The impedance of the tank circuit is 


Xcp (X. 4 - Xc.) 

Re ~ gm Xcp Xc. 


1—289 (2) 


This equation, rather than equation (1) is the real 
equivalent of equation 1 — 284 (1) and serves the 
same purpose in that it defines the conditions 


Zl = 


Z 


R. 


7 

pgc 

+ Zp 

- -h jXc 


p 


(Re ~ jXcp) jXcp 
Rp jXcp *4“ JXcp 


The real component, XcpVRe, is equivalent to Rzp, 


WADC TR 56-156 


133 



Section I 

Crystal Oscillators 


and the imaginary component, jXcp, is equivalent 
to Xz,,. Thus, the tank circuit does not appear as a 
pure resistance in the limit, but approaches an 
effective 

Q = |Xcp I R. 

Rz. XcpVRe I Xcp 1 

Re 

I X, - Xc, I Q, 

This is the same effective Q. as was determined 
qualitatively from the viewpoint of phase angles. 
The term ReXc„/Rp in equation (3) is therefore 
not to be interpreted rigorously as the total dy- 
namic reactance, but as that part of the dynamic 
reactance that is a function of the tube param- 
eters. At all times, the total l(^d reactance across 
the crystal terminals is X, = — I/mC, = Xc* + Xcp' 
which is always slightly greater than (Xoe -j- Xcp). 

CAPACITIVE ELEMENTS IN DESIGN OF 
PIERCE OSCILLATOR 

1-290. It was declared that equation 1 — 289 (3) 
defined the frequency requirements of a Pierce 


oscillator. Since the frequency and the value of 
X, and Xj are effectively predetermined constants, 
the primary problem involved in the solution of 
equation 1 — 289 (3) lies in determining the values 
of the lumped capacitances that must be inserted 
in the circuit to provide the correct value of Ci. 
In the average Pierce circuit the dynamic capa- 
citance can be considered negligible when com- 
pared with the total load capacitance, so that Cx 
can be assumed to equal Cp and Cg in series to a 
first approximation. With Cp and C* decided upon, 
approximate values of gm can be had by equating 
the denominator of equation 1 — 289 (2) to zero 
and solving for the transconductance. The maxi- 
mum and minimum equilibrium values of gm coin- 
cide with the maximum and minimum expected 
values of Rp, respectively. Next, assuming that 
gm Ip/Eg, and that E„ = IgXcg, where I* is the 
crystal current, a vacuum-tube and plate voltage 
are chosen which will provide a maximum Rp, but 
which will not cause a crystal unit of any expected 
Re to be overdriven. To determine the lumped ca- 
pacitances that must be added to provide the cor- 
rect values of Cp and Cg, it is first necessary to 



(A) 


C9 



Figure 7-124. (A) Conventional Pierce oscillator and butler-amplifier circuit. (B) Static capacitances of circuit 
(A). Cp = interelectrode capacitances. C„ = distributed capacitances of wires 


WAOC TR 56-156 


134 


Section I 
Crystal Oscillators 


know the values of the stray static capacitances 
in the circuit. These stray elements effectively 
create a lower limit to the capacitance across the 
crystal unit. Obviously, they must not be allowed 
to exceed the total specified load capacitance. Pref- 
erably, they should be as small as possible and not 
be so distributed that an optimum capacitance 
ratio, Cg/Cp, cannot be achieved. The static capaci- 
tances to be considered in a conventional Pierce 
oscillator are illustiated in figure 1-124. Before the 
optimum values of 0, and Ci can be determined, an 
experimental circuit should be constructed with the 
various leads and circuit components reasonable 
facsimiles of those intended in the final production 
models. With C, and C3 omitted, the static ca- 
pacitances between plate and grid, plate and cath- 
ode, and grid and cathode can be measured. C, 
and C3 can then be computed to provide the total 
required load capacitance. 

1-291. In the early days of crystal oscillators, and 
even today where the oscillator is not required' to 
meet a frequency tolerance less the 0.02 percent, 
the choice of grid and plate circuit capacitances 
was largely a matter of trial and error. Usually, 



(Ai 



(Bl 

figun I- 1 25. Variation of activitY and froquoncy in 
Piorco o$€lilator at plate-circuit capacitance it 
increated 


the final choice was based upon the combination 
that provided the maximum activity for a given 
d-c plate voltage. For example, if the grid-to- 
cathode capacitance, C*, of a Pierce oscillator is 
held constant while the plate load capacitance is 
varied from a minimum to a maximum, an activity 
curve, as measured by the gridleak current, is 
obtained similar to the one illustrated in figure 
1-125 (A). Formerly, it was not unusual for the 
optimum capacitance to be considered a value 
slightly greater than that at which the activity is 
a maximum. In military equipment the principal 
consideration now is to ensure a given total load 
capacitance. As shown in figure 1-125 (B), in- 
creasing the plate-circuit capacitance causes the 
frequency to decrease. As the frequency decreases, 
so does X, of the crystal unit, and at only one point 
along the curve will the crystal unit be operating 
into its rated capacitance. As stated previously, 
the circuit must provide the specified capacitance 
if there is to be an assurance that the required fre- 
quency tolerance is met when one crystal unit is 
replaced by another of the same type and nominal 
frequency. In an exceptional case, the most im- 
portant consideration may be to maintain a fixed 
frequency relative to some frequency standard. 
For this purpose small variations in the load ca- 
pacitance that can be made manually should be 
possible, but care should be taken that an operator 
is not to be able to vary the total more than is just 
sufficient to allow for a frequency variation equal 
to the bandwidth of the tolerance range. Other- 
wise, there can be a risk of overdriving a crystal 
unit, or of continuing in operation a defective crys- 
tal or other circuit component that should be re- 
placed before a complete breakdown is threatened. 

Measurement of Stray Capacitances 
In Pierce Circuit 

1-292. In order to measure the stray static ca- 
pacitances in a circuit such as that shown in figure 
1-124, the lumped grid and plate capacitances, Ci 
and C3, as well as the crystal, should first be re- 
moved. The remaining elements can be assumed to 
form a three-element network as shown in figure 
1-126. If an r-f choke is connected in the circuit, 
the frequency of the Q meter should be approxi- 
mately the operating frequency of the oscillator. 
The measurements are made with all vacuum-tube 
voltages off. The three capacitances in figure 1-126 
represent three independent variables, so that a 
minimum of three measurements is required to 
determine their values. A fourth measurement is 
desirable as a check on the accuracy of the first 
three. Any combination of measurements can be 


WAOC TR 56-156 


135 


Section I 

Crystal Oscillators 

p 



o 

© 


CpG *Cpg + 
Cqc *Cqc + 


C9C Cpc 
Cgc-fCpc 
Cpg Cpc 
Cp9 4 - Cpc 


© 


Cpc *Cpc + 


Cpg Cpc 
Cp9*fCgc 


* (4-9) CxY ’C«» + (Cyl OR C«ll 


Figure 1-126. Equivalent network Formed by the stray 
static capacitances of a vacuum-tube circuit. A mini- 
mum of three different measurements is required. 
The individual capacitances can be determined by 
solving simultaneously any three of the nine equa- 
tions above when the respective terminal capacitances 
are known 

* Generalized equation for a measured terminal 
capacitance (Cxr = Cpr„ Cor, or Cpr) when either 
Cry or Cyz, one of the two respective series-branch 
capacitances, is shotted out. 

made, the only restriction is that no two meas- 
urements are the same and that each of the 
capacitances is involved in at least one of the 
measurements. Three different measurements are 
possible between any one pair of electrodes. Thus, 
in measuring the capacitance between grid and 
cathode, the circuit is unchanged for the first meas- 
urement, the plate can be grounded for the second 
measurement, and shorted to the grid for the third. 
For the three measurements : 

(1st) Cgc = C*e -F 

(2nd) Cgc = Cgc -|- Cpg 
(3rd) Cgc = Cgc -f- Cpc 

Theoretically, these three measurements could be 
sufficient, but one or more additional measure- 
ments are needed as a check in the event that the 
Q-meter leads or the shorting wires have signifi- 
cantly affected the readings. With Cp,, C,,,, and Cgc 
determined, the lumped capacitances for both grid 
and plate circuits can readily be determined. 


BIAS VOLTAGE OF PIERCE CIRCUIT 


1-293. Oscillator bias voltages are not as critical 
as are those of other types of vacuum-tube circuits, 
for usually it is neither wave shape, maximum out- 
put, nor maximum power efficiency that is of most 
importance, but simply a constant fundamental 
frequency. For this purpose, the Pierce circuit can 
be designed to obtain a maximum Rp with a mini- 
mum drive level, and the optimum bias voltage 
will be the one that most nearly answers the need. 
As a general rule, in the case of a vacuum tube that 
is not being driven to saturation, the effective Rp 
increases as the bias becomes more negative. If 
the bias is sufficient for the tube to be operated 
class B or class C, so that the plate current is cut off 
for an appreciable part of each cycle, such oper- 
ation greatly improves the power efficiency. How- 
ever, a bias developed across a gridieak resistor 
can never exceed the peak excitation voltage dur- 
ing stable oscillations. The excitation voltage, in 
turn, is limited by the current Ig, which must not 
exceed a value that would cause the losses in the 
crystal to exceed the permissible limit. When i , 
is maximum, the limiting crest value of Ig is 


Igm 



, where P™ 


is the maximum permis- 


sible drive level in watts, and R^m is the specified 
maximum effective resistance when the crystal 
unit is operating into its rated load capacitance. 
The maximum grid bias is thus 


(max) Ey = - 1-293 (1) 

\ ^ein 

where Zg ^ is the r-f impedance between 

grid and cathode. It was found earlier (paragraph 
1-284, equation 1 — 284 (8) ) that a maximum Rp is 

to be had when Cg = — ''.IC . or approximately 

when Cg = Cp 2Ci. Equation (1) under these 
conditions becomes 

(max) Ee = - , 1-293 (2) 

2uC,VRy,„ 

Equation (2) is quite important in that it shows 
that the maximum grid bias obtainable with a 
maximum R^ is fixed by the crystal specifications. 
The maximum E.. for any given crystal unit may 
be obtained from equation (2) by substituting the 
actual Re for R,.„,. The proper anode voltages for a 
given vacuum tube, or vice versa, to provide a re- 
quired output are consequently also effectively 
predetermined by the crystal specifications. For 
maximum stability with any given vacuum tube in 


WADC TR 56-156 


136 



Section I 
Crystal Osciiialors 


a conventional Pierce circuit, C( and Cp should 
each be made equal to 2Ci, or if the amplification 
factor is small, C,/Cp should be made equal to 

- ^ with the total capacitance in series made to 

equal C^. With the capacitance so determined, the 
plate and screen voltages can be adjusted to give 
the desired output and excitation voltages. Since 
Ep Ep, the output voltage is also effectively lim- 
ited by the crystal specifications. With Cg = Cp = 
2Ci, (max) Ep = .707 (max) |E,.|, where (max) Ep 
is the maximum r-ih-s value of Ep, and (max) E^ is 
given by equation (2). 

FIXED BIAS FOR PIERCE OSCILLATOR 

1-294. It is not conventional to employ a fixed bias 
in a crystal oscillator, although it can be done — 
even to advantage in some cases. An r-f choke can 
be substituted for the gridleak resistance, thereby 
reducing the grid losses to an absolute minimum as 
long as the excitation is insufficient to overcome 
the bias and cause grid current to flow. In order 
for stable oscillations to be maintained, an increase 
in excitation roust cause a decrease in amplifica- 
tion, and a decrease in excitation must cause an 
increase in amplification. When using a fixed bias, 
the choice of operating voltages is much more re- 
stricted than when employing gridleak limiting. 
Because of the more critical operating conditions, 
the replacement of one crystal unit with another 
having a different resistance may require addi- 
tional circuit adjustments. If a fixed, class-B or 
class-C bias is used, a slight decrease in the ampli- 
tude of oscillations normally leads to the oscilla- 
tions dying out all together. This is because the 
average amplification of the positive alternation 
of each cycle increases and decreases directly with 
the amplitude instead of inversely. For instance, 
with a class-C fixed bias, a decrease in the ampli- 
tude of one cycle would mean that the tube is cut 
off during a larger fraction of the succeeding cycle, 
thereby further decreasing the average amplifica- 
tion. On the other hand, if oscillations were once 
started, the tendency would be for the amplitude 
to build up until limited by grid and plate satura- 
tion. Only if limiting is provided by nonlinear ele- 
ments, such as thermistors or varistors, in the 
external circuits is class-B or class-C fixed-bias 
operation possible if the tube itself is not to provide 
the limiting action. A fixed bias can be used if the 
tube is driven to saturation each cycle, but such 
operation is not practicable unless the utmost 
power is required from the oscillator, and in any 
event should not be attempted with the Pierce 
circuit. The only feasible application of a fixed bias 


ib 



Figure 1-127. Point on It curve at which stable 
oscillations can be achieved with fixed bias. For 
minimum grid losses, the peak of the maximum 
excitation voltage must not exceed the fixed 

in the Pierce oscillator is to operate the tube just 
above the knee of the E,.Ib curve (see figure 1-127) . 
(E,. and Ib represent no-signal, d-c values of grid 
voltage and plate current respectively.) At the 
indicated operating point a slight decrease in the 
activity results in the average amplification of the 
negative alternation of the excitation cycle being 
greater, whereas the amplification of the positive 
alternation remains essentially constant. Thus, the 
over-all amplification of the weaker cycle is greater 
than that of the stronger, and a stable equilibrium 
is possible. An r-f choke would normally replace 
the gridleak resistor. Such operation practically 
eliminates the grid losses as long as the peak exci- 
tation voltage does not exceed the bias. Theo- 
retically, then, the fixed bias permits a maximum 
Q in the feed-back circuit, and in this respect aids 
the frequency stability. There are, however, prac- 
tical difficulties involved. If the C«/C,, ratio of the 
external circuit is such that the oscillator is to 
operate at the minimum g„„ it may be difficult to 
find a vacuum tube that provides the desired trans- 
conductance when operated at suitable voltages 
just above the knee of the EJb curve. The low 
transconductance can readily be achieved by using 
a remote-cutoff tube, but the amplitude stability 
will be more critical since the amplification of the 
positive alternations increases with amplitude, 
thereby tending to annul the limiting action of the 
negative alternations. Variations in the circuit 
capacitances have the following effects, which are 
essentially the same as those that occur with grid- 
leak bias except that the amplitude variations are 
more pronounced in the fixed-bias circuit. A small 
decrease in the capacitance ratio, say by an increase 
in C,„ would mean that the voltage ratio, 

Ep/E* = C,/Cp = = g,.. Z,, 


WADC TR 56-156 


137 



Section I 

Crystal Oscillators 

must become smaller. Since Ep/'Eg varies inversely 
with Cp whereas Z,, varies inversely with C,,', an 
increase in C,,, as long as Rp>> Z),, requires gm to 
increase proportionately in order that equilibrium 
may be re-established. In other words, an increase 
in C|, must cause the excitation voltage to decrease. 
Conversely, a decrease in C,, causes g„, to decrease 
and the excitation voltage to increase. Rp varies in 
a direction opposite to that of g„„ but the per- 
centage change in Rp is not as large as the per- 
centage change in g,„. The change in g,„ is approxi- 
mately proportional to the change in Cp. If Cp is 
constant, but C,. is varied slightly, Zi. will be ap- 
proximately constant, so that g„, must also vary 
directly with Cp. in order for equilibrium to be 
maintained. Therefore, the changes in the excita- 
tion voltage with changes in C^ are similar to those 
with changes in C,,. When Cp and Cg are varied 
independently the total load capacitance Cj 
changes, and hence the frequency and X, also 
change. If, during tuning adjustments, Cp and Cg 
are varied so that the same total capacitance is 
always maintained, then, in the region where Cp 
^ Cg the change in Cp is approximately equal but 
opposite to the change in Cg. In this case g„„ which 
has reached its minimum value, tends to remain 
constant, as do also the excitation voltage and the 
plate resistance of the tube. It should be under- 
stood that this optimum condition holds only for 
variations in capacitance that leave the total load 
capacitance unchanged. With Cg fixed, g„, must in- 
crease as the capacitance ratio is varied to either 
s.de of its optimum value. When Cg/Cp is made 
greater than one the change in g„, and the excita- 
tion is greater than when Cg/Cp is made less than 
one by an equivalent proportion, (e.g., Agn, is 
greater when Cg/Cp is changed from 1 to 2 than 
when it is changed from 1 to 1 / 2 -) An increase in 
g,„ can occur only by virtue of a decrease in excita- 
tion voltage. Thus, if the grid impedance, Zg, is 
made larger than the plate impedance, Zp, the crys- 
tal current, Ig, must necessarily become smaller. 
Oscillations can thus be maintained with a smaller 
drive level. Nevertheless, optimum stability gen- 
erally requires a maximum Rp, which, in turn, coin- 
cides with a maximum excitation voltage and 
minimum g,„. 

1-295. According to equation 1-284(10), the opti- 
mum capacitance ratio will automatically cause 
the oscillator to seek an equilibrium when Rp = 

PI 

(/t -f- 1) Zi„ or approximately when Rp = 
where PI is the performance index. Herein lies the 
principal limitation of fixed-bias operation of a 
Pierce circuit. The maximum to minimum values of 


the PI of a given type of crystal unit can be as 
much as 9 to 1 for the same frequency — and 
greater still if the oscillator is to operate over a 
wide frequency range. It may be difficult to find 
a tube to provide equivalent variations in the effec- 
tive Rp unless the excitation voltage is to be so 
large that it drives the grid positive on the positive 
excitation peaks. In this case, the principal advan- 
tage of the fixed-bias— to maintain a maximum Qf 
and to minimize the variations in the input imped- 
ance — is lost. Since an increase in Rp must be 
accompanied by an increase in the r-f plate current 
operating into a proportionally increased load im- 
pedance, the tendency will be for the crystal power 
(approximately equal to Ip* Z|,) to vary directly 


with (PI)* or^^^ . Unless there is some guaran- 
tee that the maximum Rg is not to be greater than 
twice the minimum expected value, some additional 
form of limiting must be used in the fixed-bias 
circuit, such as connecting a varistor across the 
r-f load, to ensure that the low-resistance crystal 
units are not over-driven. The fixed bias should not 
be less than three times that given by equation 
1 — 293(2). The vacuum tube (preferably a pentode, 
because of its low plate-to-grid capacitance and 
high Rp) should be chosen and the anode voltage 
determined that permits operation at the lower end 
of the straight portion of the E,.Ib curve. The opti- 
mum bias and plate voltages are best established 
by experiment. The principal problem is to ensure 
a sufficient output for crystal units of maximum 
Rg, without overdriving those crystals of minimum 
Rg. The average crystal unit has an Rg approxi- 
mately one-third the maximum. An occasional crys- 
tal unit may have a value of Rg perhaps as small 
as one-tenth the maximum. It can be seen that a 
serious obstacle to the use of a fixed bias is that 
manual adjustments of the operating voltages are 
necessary when replacing crystal units, unless the 
plate circuit is to be rather heavily loaded. The 
output will tend to vary by a large factor from one 
crystal unit to the next. The same problem is en- 
countered with the use of a gridleak bias, but volt- 
age adjustments are not absolutely necessary, even 
under no-load conditions. A familiarity with fixed- 
bias operation is helpful, however, in that it aids 
the understanding of gridleak operation. 


GRIDLEAK BIAS FOR PIERCE OSCILLATOR 

1-296. The importance of having gridleak instead 
of fixed bias is two-fold: First, it permits a large 
initial surge of plate current, so that oscillations 
will build up quickly. If the tube were being oper- 
ated class B or class C with a fixed bias, the bias 


WADC TR 56-156 


138 



would have to be removed before oscillations could 
build up at all. Second, it ensures a maximum sta- 
bility in the output. If for any j-eason the excita- 
tion should increase or decrease, the d-c grid 
current and hence the bias follows the change, 
always acting in a direction that tends to annul 
the original change. When the oscillator is first 
turned on, the starting bias is zero regardless of 
the value of the gridleak resistance. Thus, insofar 
as the initial surge of current is concerned, the 
value of the gridleak resistance, R„, is not a first- 
order factor. However, the value of is significant 
in its effect upon the total build-up time. This effect 
is considered in paragraphs 1-304 and 1-305, where 
the conditions most favorable for oscillator keying 
are discussed. The present discussion considers 
only the effects of R^ upon the oscillator stability 
after the oscillations have reached a maximum am- 
plitude. First, it is desirable that the grid losses 
be as small as possible, and that they at least can 
be considered negligible by comparison with the 
losses in the crystal unit. For a continuous flow of 
d-c grid current to be maintained, the grid must be 
positive with respect to the cathode at the positive 
peak of each excitation cycle. The amount of grid 
power that is dissipated, the extent to which the 
grid becomes positive, and the length of the period 
during which the grid is positive and electrons are 
flowing from cathode-to-grid depend upon how 
great a percentage of the total charge escapes 
through Re during the remainder of the cycle. This, 
in turn, depends upon the ratio of the period of one 
cycle to the RC time constant of the grid circuit. 

This ratio, l/R„Cgf, is seen to be equal to ^”’ 

Kg 

The smaller this ratio can be made, the smaller will 
be the percentage leakage of charge during one 
cycle, and the more nearly will the bias remain con- 
stant and equal to the peak excitation voltage. At 
high frequencies, ratios on the order of 1/50 and 
smaller are quite easily obtained. With the period 
of one cycle so short compared with the time it 
would take 63 per cent of the accumulated charge 
to leak off, it can be assumed that the bias voltage 
equals the peak excitation voltage in magnitude. 
Should the excitation voltage increase, the bias also 
increases. The peak excitation voltage is 

E,„, = Ie„, Ze = 1.414 Ie Ze = I Ee 1 

1—296 (1) 

where Eo is the grid bias when ReCj > > 1/f, and 
Ig,„ is the peak r-f grid-circuit current. The bias 
voltage is also given by the equation 

Ee = Rg 1—296 (2) 


Section I 
Crystal Oscillators 

where Ic is the d-c grid current. If an r-f choke 
having an impedance that is large compared with 
Rg is connected in series with R^, the r-f voltage 
across Rg becomes small compared with the d-c 
voltage. With this arrangement it is readily seen 
that the approximate grid power expenditure is 

Pg = Ic E„ = EcVR* 1-296 (3) 

If the r-f choke is not present, so that the voltage 
across Rg varies sinusoidally from a peak of — 2Eg„ 
to a peak of 0 on the positive alternation, the aver- 
age squared voltage across Rg, equal to 

^ f (1 -f sin «t)'‘ d(a)t), is found to be 1.5 Ec*. 
Thus, in the absence of an r-f choke, 

Pg = 1.5 EcVR* 1—296 (4) 

Clearly, if the grid losses are to be held to a mini- 
mum, Rg must be as large as possible. If an exami- 
nation is made of several representative crystal 
oscillator circuits in actual production, it will be 
discovered that very few employ gridleak resist- 
ances higher than 100 kilohms, and only an occa- 
sional value of Rg is found higher than 0.5 meg- 
ohms. The answer is principally to be found in the 
fact that the oscillator design is usually a com- 
promise among several factors: (a) frequency 
stability, (b) output-voltage stability, (c) output 
control, (d) operating efficiency, (e) maximum 
economy in production costs, (f) minimum over-all 
weight and space requirements, (g) whether or not 
oscillator is to be keyed, (h) frequency range (i), 
value of Cg, (j ) whether or not circuit is to pennit 
switching from crystal to tuned circuit, and (k) 
the suppression of parasitic frequencies. Either a 
high or a low value of Rg can improve the per- 
formance in respect to any one of the factors above, 
depending upon what is required concerning the 
other factors. For example, a very large Rg can 
improve the frequency stability by reducing the 
grid losses, when only a small output is required. 
On the other hand, the. same value of Rg could lead 
to both frequency and output voltage instability 
if maximum output or maximum operating effi- 
ciency were required. The effects of Rg relative to 
various factors listed above are discussed briefly 
in the following paragraphs. 

Gridleak Resistance and Frequency 
Stability of Pierce Oscillator 

1-297. The grid Rg can lead to frequency instability 
in two ways. As Rg is decreased, the grid losses load 


WADC TR 56-156 


139 



Section I 

Crystal Oscillators 

the feed-back circuit and require that the tank be 
operated farther away from resonance. Rp of the 
vacuum tube is reduced and its effect upon the 
frequency becomes more pronounced. On the other 
hand, if is increased indefinitely, the oscillator 
can become self-modulated. This latter effect is 
more properly classed as a problem of output volt- 
age stability, and is discussed in the paragraph 
1-299. The principal frequency-stability importance 
of Rg is the degree by which it reduces the effective 
Qf of the feed-back arm. (Qf is the feed-back qual- 
ity factor only from the point of view of phase rota- 
tion.) As defined by equation 1 — 272 (1) 


current pulse when the grid is positive with respect 
to the cathode will be considered negligible com- 
pared with the power losses in Rg per cycle. Assum- 
ing that no r-f choke is used and that Rg is 
sufficiently large so that equation 1 — 296 (4) is 
approximately correct, equation (1) on substitu- 
tion and rearrangement becomes 

R*' = 1-297 (2) 

Since it has been assumed that Rg is large com- 
pared with Xcg, then Xcg' <=■ Xo* and Eg* = 

2(IgXcg)’. On substitution in equation (2), 


Q _ Qpirc Qk 

' Qptc + Q* 

where Q,,gg is the actual over-all effective Q of the 
feed-back arm, and Qg is the effective Q of the in- 
put circuit. In order to analyze the effect of Rg in 
terms of the equivalent Pierce r-f circuit, we repre- 
sent Zg by an equivalent reactance, Xcg' in series 
with an equivalent resistance, Rg', as shown in fig- 
ure 1-128. The problem now is to determine Rg', 
such that 


Ig^ Rg' = Pg 


1—297 (1) 


where Pg is the power expended in the gridleak 
resistance. The power expended during the short 



CD ^pgc — 

© Og = 

Figure 1-128. 


X a X C q 
R« + R’g 


© Q» = 




R'a 


Qpqc Qg 

OpgC + 0 g 


fquiVa/enf Pierce feecf-bacfc circuit 


WADC TR 56-156 


3Xo,“ 


1—297 (3) 


If an r-f choke were used in series with Rg, the 
approximate value of Rg' would be 


c» 


R. 

The values of Qg become 

(without choke) Qg = 


I Xcg' I 

R*' 


1—297 (4) 


Rg 


3 Xcg 
1—297 (6) 


(with choke) 


Qg = 


Rg 


2X, 


Cg 


1—297 (6) 


The equations for Qpgg are 


/ -il. i I. 1 \ /-V (Xe + Xcg) Rg 

(without choke) Qpgg = p p , o v 2 

Kg K, -|- o Acg 

1—297 (7) 

. -iL 1. 1 ^ (Xe + Xcg )Rg 

(With choke) Qpgc = p p 1 9 y * 

XVg IVe “T ^ 

1—297 (8) 


Assuming that Qpgg Qg > > 1, Q( is given by the ap- 
proximate equations 

(without choke) Qi = - 


(with choke) 


Re 


3 X. Xcg 
Rg 


Qf 


1—297 (9) 

(Xe + Xcg) 


Re - 


2 Xe Xc, 

Rg 


1—297 (10) 


140 



In the previous discussions it has been supposed 
that the grid losses are kept negligible compared 
with the losses in the crystal, so that a Qr equal to 


X, + Xo 

K. 


* is approximately correct. Where this 


assumption cannot be made, those equations that 
define the frequency of the oscillator, such as equa- 
tion 1 — 289 (3) , can still be used for approximately 
correct answers if R, is replaced by the appropriate 
denominator in equation (9) or (10). Thus, the 
effective frequency-determining resistance of the 
feed-back arm, can be defined as 


R,. = R.-l-|3X.Xc,/R,| or R, -h 1 2 X, Xc,/R. | 

1—297 (11) 

On the other hand, in those equations that concern 
the equilibrium between the energy input and out- 
put of the feed-back arm, such as equation 1 — 289 
(2), the effective feedback-circuit resistance to 
substitute in the place of R, is the value 


R( = Re -|- Rg' 


1—297 (12) 


On multiplying equation (11) by Rg/R*. it is ap- 
parent that if Rg is to have a negligible effect upon 
the frequency, it must be much greater than 

^ • Similarly, if Rg is to have a negligible 

xV* I 

effect on the total feed-back power requirements, 
according to equations (3) and (12), it must be 
much greater than 3Xog*/Re. If the oscillator is to 
be operated in the region of maximum R,, of the 
tube, |Xcg| will approximately equal X,/2, and 


3X,X, 


R. 


££. 


will approximately equal g PI. Under 


these conditions, a good rule of thumb, if the other 
operating requirements permit, is to employ a grid 
resistance equal to 15 times the minimum permis- 


sible PI ' o T c * R ')* greater, where Re„, = 
(max) Rg. 


Grid~Resistance Effects and “Class-D” 

Operation of Pierce Circuit 

1-298. Typical curves showing the effect of the 
grid resistance upon the frequency and the fre- 
quency stability of a Pierce oscillator are shown 
in figure 1-129. The curves in figure 1-129 (A) were 
obtained for plate voltages (Eb) of 50, 60, and 70 
volts under no-load conditions ; the curves in figure 
1-129 (B) were obtained for plate voltages of 75, 
100, and 125 volts when a load resistance of 5000 
ohms was connected across the plate circuit. The 
curves were plotted from measurements of an ex- 
perimental Pierce oscillator during a USAF re- 


section I 
Crystal Oscillators 

search project under the direction of E. Roberts at 
the Armour Research Foundation, Illinois Institute 
of Technology. The frequency deviation due to a 
change in R„ with a change in plate voltage is in- 
dicated by the frequency difference between the 
points on the curves that correspond to the same 
values of grid resistance. The four sets of curves 
for each of the two load conditions were obtained 
by maintaining the grid-to-cathode capacitance 
constant and varying the plate-to-cathode capaci- 
tance. The bottom set of curves in each graph 
represents the closest approach of the four sets to 
a Cg/Cp ratio of 1, and the top set represents ap- 
proximately the largest Cg/Cp ratio (maximum Ep) 
at which oscillations can be maintained. As is to be 
expected the frequency deviation due to changes 
in voltage is greater for the larger than for the 
smaller values of C,/C„. However, part of this 
greater deviation is due to the fact that in the cir- 
cuit from which these curves were plotted, the 
plate capacitance was obtained from a capacitor 
paralleled by an inductor. At the minimum effective 
value of Cp, the capacitor and inductor approach 
a state of parallel resonance, so that the rate of 
change in the equivalent plate reactance with a 
change in Rp is larger than would be the case if 
no inductor were present. Also, the fact that the 
total load capacitance facing the crystal is smaller 
at the higher frequencies contributes to the fre- 
quency deviation. Partially counteracting this lat- 
ter condition is the increase in the effective Qp of 
the crystal and in the Qr of the feed-back circuit as 
a whole because of the rise in frequency. There are 
three variables influencing the frequency that are 
affected by a change in the grid resistance. The 
first of these is the Q* of the grid-to-cathode im- 
pedance. As Rg becomes relatively small, Q, be- 
comes the dominant factor determining the phase 
of Ep with respect to Eg. As Qg decreases, the tank 
circuit must appear more capacitive. Thus, the re- 
actance of the crystal unit in the feed-back arm, 
and hence the frequency, must increase. This is 
indicated by the sharply rising tails of the curves in 
figure 1-129. The second frequency-determining 
factor affected by Rg is the effective grid-to-cathode 
capacitance, Cg. As Rg' decreases, Cj effectively in- 
creases. That is, if Rg and Cg are represented by an 
equivalent Rg' and Cg' in series, it can be shown 
that Cg' increases with a decrease in R*. As a result 
of this effect, as Rg becomes small, the frequency 
tends to decrease; but, as indicated in the upper 
sets of frequency stability curves of figure 1-129, 
when Cp is small compared with Cg, changes in Cg 
have little effect since the total load capacitance is 
approximately equal to the smaller capacitance. On 


WADC TR 56-156 


141 



Section I 

Crystal Oscillators 

the other hand, when the Cg/C,. ratio is small, as 
indicated by the lower-frequency curves in the fig- 
ure, the changes in the effective Cg have a measur- 
able effect upon the frequency. It can be seen at 
the low values of grid resistance that the effective 
increase in capacitance greatly diminishes the rise 
in frequency that would otherwise occur because of 
the decrease of the grid-to-cathode, Qg. Indeed, the 
bottom curves in figure 1-129 (A) show that for 
the particular circuit and crystal unit the two op- 
posing frequency effects of R, apparently cancel 
each other when Rg is in the neighborhood of 100,- 
000 ohms. This same set of curves indicates that 
a minimum frequency deviation with plate voltage 
occurs when the grid resistance is approximately 
200,000 ohms. It is with some diffidence that we 
attempt to explain the reason why this particular 


value of Rg should provide a point of maximum 
frequency stability. Rather than interpret the 
effect as due to a possible optimum ratio, or as due 
to a possible variation of Rg with plate voltage that 
tends to cancel the effect of the variation in Rp, it 
seems more likely that the optimum results are 
due to a coincidence between the third frequency 
factor mentioned above and the characteristics of 
the tube, a 6C4 (triode), that was used in the test 
circuit. This third frequency factor is the average 
grid bias, which tends to increase and decrease 
with Rg, although the variations are not pro- 
nounced when Rg is large. As the bias increases, 
so also does Rp, which in turn causes the frequency 
to decrease. This can best be seen from an examina- 
tion of the curves at the top of figure 1-129, where 
the effects of Rg on Cg' are negligible as they affect 



Figure 1-129. Frequency of tuned Pierce oscillator versus plate voltage and grid resistance for various plate- 
tuned load capacitances. (Max) represents plate tuning adjustment that provided maximum r-f plate voltage 
when grid resistance and d-c plate voltage were values indicated by zero reference point. A 7-mc CR-I8/U 
crystal unit was used, having a PI of 49,000 ohms when operating into a rated load capacitance of 32 /ifJ 


WADC TR 56-T56 


142 



Section I 
Crystal Oscillators 


the frequency. As R* increases, the bias approaches 
as a limit the magnitude of the peak excitation 
voltage. Thus, R„ also rises to some limiting value, 
causing the frequency to level off to some minimum 
value. If the plate voltage is increased when the 
tube is biased below the straight portion of the 
Eelb curve, one result is a decrease in R,„ which, in 
the curves of figure 1-129, clearly causes the fre- 
quency to increase. However, an increase in the 
plate voltage also causes an increase in the excita- 
tion voltage, and hence in the grid bias. Thus, it 
may well be that the point of maximum frequency 
stability as indicated in the bottom curves of figure 
1-129 (A) is the result of an increase in bias with 
the increase in plate voltage just sufficient to hold 
R,. constant. A class of operation such that the 
percentage change in R„ due to a change in Eb is 
annulled by a percentage change in Rp due to a 
change in E,., or vice versa, suggests interesting 
possibilities in stabilizing the plate resistance by 
methods other than plate-supply regulation. There 
is no evidence that this type of operation has been 
investigated, but on the strength that possibilities 
exist for practical application in oscillator circuits 
not employing a fixed bias, the name "class-D” 


operation is proposed. Three subclasses are pos- 
sible : “D„” where changes in Ei, can occur more or 
less independently and E,. is the dependent vari- 
able; “Dj,” where E.. can be considered independent 
and Eb dependent; and “D, " where E,, and E,, are 
mutually dependent. Mathematically, this class of 
operation can be defined by the following equa- 


tions : 



(Class D) 

ARp 1 /0R,.\ 
Rp Rp \ 9 Eb / 

AEb 

£ = const. 

c 


^ 1 
b 

= 0 
const. 


4E. - AE. 

1—298 (1) 

(Class D.) 



(AE^ independent) 

D 



ae.-|5-ae. 

1—298(2) 

(Class Dj) 



(AE^. independent) 

1—298 (3) 

(Class D,. s) 

Equations (2) and (3) both apply. 


(B) 

(= 7MC 



WADC TR 56-156 


143 



Section I 

Crystal Oscillators 

Equations (1), (2), and (3) sufficiently define 
“class-D” operation, but the parameter, R,„ repre- 
sents a with-signal average plate resistance, and 
not the instantaneous or static resistances repre- 
sented by conventional R„ curves; although the 
static values would apply if the signal amplitude 
were small relative to the bias, as would be the case 
if age were being used. An increase in plate voltage 
can generally be expected to cause an increase in 
excitation and grid bias in the conventional oscil- 
lator circuits. This action, in turn, can be expected 
to cause an increase in the average R,,. Thus, if 
“class D” is to be in effect, it is necessary that an 
original increase in cause R,, to decrease by the 
same amount as the change in E,. will cause it to 
increase. For this to occur, the plate characteristic 
curves must show positive slopes that increase with 
plate voltage; that is, the Ei>Ib curves must be 
curving upward in the direction of increasing E,, 
at the operating voltage, as is quite characteristic 
of triode curves. Pentodes do not show this char- 
acteristic if the screen voltage is held constant, 
since the plate resistance tends to increase with 
increasing plate voltage. However, if the screen 
voltage varies with, and in the same direction as, 
the plate voltage, as can be the case when the two 
voltages are obtained from the same source, plate 
characteristics can be achieved similar to those of 
triodes, but with the advantages of larger values 
of R,. and an independent variable (E. j) by which 
the rate of change of R,, with Eb, ( 9R,./9Eb) , can be 
adjusted. Now there is an additional implied con- 
dition that must be met if class-D effects are to be 
achieved in conventional oscillator circuits. This 
is the requirement that g„, also remain constant. 
For example, by equation 1 — 289 (2) which is re- 
peated here 

o _ Xco (X„ -|- Xcr) 

” Re - g,„Xcp Xc* 

it can be seen that feed-back equilibrium in a 
Pierce circuit requires that as long as R„ and the 
external circuit parameters remain constant, so 
also must g„,. An analogy here is to be found in 
class-A operation, which is defined by the operation 
of the tube along the straight portion of the E,.I|, 
curve, i.e., in a region of constant g„,. Since the 
principal purpose of class-A amplification is a dis- 
tortionless output, by implication a necessary re- 
quirement is that the operation also be in a region 
of constant R,,. In “class D,” on the other hand, a 
constant effective R,, is the sufficient definitive con- 
dition, but in application a constant effective gm is a 
necessary implication. If g„, is to remain con- 


stant, equations for Ag,„ similar to (1), (2), and 
(3) for aRp must hold simultaneously. This can 
be achieved in one of two ways, or a combina- 
tion thereof. Assume that the plate voltage in- 
creases. If Rg is sufficiently small for the positive 
excitation peak to drive the grid reasonably far 
above the zero point, an increase in excitation, al- 
though decreasing the average g„ on the negative 
alternation, can annul this effect by increasing the 
average g^ in the region above the zero grid volt- 
age. For this to occur, the tube would have to be 
operated at plate voltages low enough for the zero 
grid point to lie well within the bend of the E,.lb 
curve. This, indeed, was the state of the 6C4 tube 
when the curves of figure 1-129 (A) were deter- 
mined. Presumably, under no-load conditions and 
a small Cg/C„ ratio, as the grid resistance was 
decreased, the changes in the positive excitation 
peaks with changes in plate voltage were just 
sufficient to maintain g„, approximately constant 
for values of Rg between lOOK and 200K. If Rg is 
very large, the effect above is negligible, since any 
increase in the positive excitation above the zero 
grid voltage point becomes minor compared with 
the total increase of the negative alternation. For 
g,„ to be stabilized when Rg is large, the change in 
plate voltage must make a change in the cutoff 
voltage comparable to the change occurring in the 
excitation voltage. Plate characteristics most prob- 
ably favorable to “class-D” operation appear to be 
had with low plate voltages. Unlike the R,, and gm 
in the conventional classes of amplifier operation, 
where R„ and g„, can be varied independently, this 
condition cannot exist in class-D operation of con- 
ventional oscillators, since R„ and gm are tied to- 
gether by the feed-back energy requirements at 
equilibrium. Any condition that would stabilize the 
one, would automatically stabilize the other. It 
is only R,„ however, that directly affects the phase 
of the feedback. Once R,, becomes large relative to 
the impedance across the tube, the percentage 
variations in g,,, become very small, so that gm can 
be assumed to be a constant for all practical pur- 
poses. It is R|, that requires critical attention if it 
is to be held constant. As discussed in paragraph 
1-342, the curves shown in figure 1-146 strongly 
suggest the possibilities of “class-D” operation in 
the case of a Miller circuit. The solution of the 
“class-D" equations for a given vacuum tube and 
circuit can probably be approximated graphically, 
using families of R„ curves versus E^ and £„ or 
curves of the deviations of the R„Eb and RpE^ 
curves. If rates of change of E,. with Ep can be ob- 
tained that provide a solution for equation (1) 
when the values of Ec and Ep are practicable, the 


WADC TR 56-156 


144 



possibility exists that an oscillator of any type, 
parallel- or series-resonant, using a gridleak bias 
(or age) can be designed so that for all practical 
purposes it is independent of small fluctuations in 
the plate-supply voltage. Empirically, “class-D” 
operation is indicated at the point or points where 
the frequency deviation curves of an oscillator 
change in sign, or where frequency curves, such 
as those in figure 1-129, cross or touch each other. 
A full analysis of this class of vacuum-tube opera- 
tion is beyond the scope of this handbook. It is 
suggested here only as a possible line of inquiry. 

Gridleak Resistance and. Output Voltage 
Stability of Pierce Circuit 

1-299. The stability of the output voltage depends 
largely upon how readily the gridleak bias can 
follow small fluctuations in the excitation voltage. 
Imagine, for example, that the vacuum tube is 
being operated class C, and that after equilibrium 
is reached the positive peak of a certain excitation 
cycle happens to be slightly higher than the aver- 
age. The average bias during the succeeding cycle 
will thus be slightly more negative than is normal, 
so that during this period the tube is conducting 
a smaller fraction of the time, and the peak excita- 
tion voltage will drive the grid less positive than 
before. This means that the amplification during 
this cycle will be less than the amplification dur- 
ing the preceding cycle. If the Rt.C„ time constant 
is extremely large compared with the period of a 
cycle, the bias remains relatively fixed for the dur- 
ation of several cycles. In which case the peak of 
several succeeding cycles must rise to progressively 
lower points on the E,.Ib curve. The oscillations will 
continue to decay until a sufficient amount of the 
bias charge of has leaked through Rg to permit 
oscillations to again build-up. For this reason Rg 
cannot be increased indefinitely without the risk 
of the oscillator becoming self-amplitude-modu- 
lated. As Rg is gradually increased, the amplitude 
sooner or later begins rising and falling at a radio- 
frequency rate. If Rg is further increased, the 
modulation of the output can fall within the audio 
range. Finally, with extremely large values of Rg, 
the circuit behaves as a damped-wave blocking 
oscillator. Now, assume that Rg is infinite. As oscil- 
lations build up, the bias for each cycle is essen- 
tially the same as the peak excitation voltage of 
the preceding cycle. Eventually a peak excitation 
voltage is attained which causes the bias for the 
next cycle to be too great for the circuit to be 
resupplied with all the energy that will be lost 
during the period of the cycle. If the peak bias is 
equal to or greater than the cutoff bias, the oscil- 


Section I 
Crystal O^illalors 


lations will die out completely, since any decrease 
in excitation with class-C bias means a decrease in 
average amplification. To avoid this possibility, it 
is important that sufficient electrons escape from 
the grid so that, at the beginning of the cycle im- 
mediately following the first peak cycle, the bias 
will have returned to approximately the same 
starting point. Expressed in another way, to avoid 
the intermittent activity, there must be an assur- 
ance that the positive peak of every cycle will 
drive the grid positive. This assurance is to be had 
for all operating conditions if the bias voltage de- 
creases at a greater rate than would the positive 
excitation voltage peaks if the tube were cut off 
for a complete cycle. In practice, the vacuum tube 
can be conducting in polar opposition to E„ and 
hence effectively supplying energy to the circuit 
during the entire negative alternation of an E,, 
cycle. Nevertheless, if there is an assurance that 
the bias voltage drops as fast as the peak excita- 
tion voltage when no energy is being supplied to 
the circuit, the bias reduction is certainly sufficient 
if the net rate of energy-loss is reduced by virtue 
of a variable release of energy by the tube through- 
out a large part of each cycle. The problem, then, 
becomes one of first determining the percentage 
change in the peak excitation voltage that would 
occur during the period of one cycle if the tube 
were suddenly cut off. 


1-300. At the instant that Ig is a maximum, the 
voltages across the reactances in the tank circuit 
are zero, and none of the circuit energy is stored 
in the capacitances. All the stored energy at that 
instant is in mechanical form, and is equal to the 
kinetic energy of the crystal as it swings through 
its position of zero potential energy. As discussed 
in paragraph 1-249, this stored energy is equal to 
I,’L, where R is the series-arm current, and L is 
the equivalent series-arm inductance. Now, when 
the crystal appears as an inductance, R is approxi- 
mately equal to R plus the current, Ro, which flows 
through the shunt capacitance, C„, of the crystal 
unit. (Only the unsigned magnitudes of R and Ic» 
are considered here.) We can say, approximately, 
that the stored energy is equal to (R -|- Rol’E- 
As is also discussed in paragraph 1-249, the ratio 
of stored energy to the energy dissipated per ra- 
dian, is equal to the Q of the circuit, which in 


this case is effectively ; b » > or the 

(Ke -|- Kg 4" Mg 

eouivalent value — 

qu aent aiue, (R/-hR,')V’ 

where RR is the equivalent load resistance when 
represented as in series with the plate-circuit ca- 
pacitance, and R is the series-arm resistance of 


WAOC TR 56-156 


145 



section I 

Crystal Oscillators 


the crystal. If R^' and Rr/ can be considered neg- 
ligible. the circuit Q will be the actual crystal 

plate current is cut 

off for an entire cycle, which is a period of 2ir 
radians. The fraction of the energy dissipated 
during this time is approximately equal to 2ir/Q = 
R/fL, if Q is of sufficient magnitude that the per- 
centage decrease in current is not large. Since the 
energy is proportional to the square of the voltage, 
the equivalent decay in peak excitation voltage is 
E„„\/R/fL |E..|\/R/^- If E,., the grid bias, 

whose magnitude only we shall consider, is to de- 
crease at the same rate, the bias charge, equal to 
C^Ee, must leak through Rj at an average rate of 
CgE.VR/fL during the period of one cycle. Thus 

(min) I, = - p = fC*E, VRTfL 
(max) Rg 

1—300 (1) 

The maximum safe value of Rg for all operating 
conditions, if Rg' and Ri/ are negligible compared 
with R,., according to equation (1) is 


(max) Rg = VL /Cg VfR 1—300 (2) 

Since VE' = l/w\/C, where C is the equivalent 

Q 

series-arm capacitance, and since C ■=« in the 

case of partially plated elements, where r is ap- 
proximately equal to the theoretical capacitance 
ratio, r,., given in figure 1-95, then, on substitution 
in equation (2) 


(max) Rg 


‘•'CgV fRc„ 


1—300 (3) 


In paragraph 1-297 it was shown that when 
Cg = 2Cx, if Rg is to be considered negligible it 
should be at least 15 times the minimum permis- 
sible PI. Assume that Rg/ (min) PI = k, (k is not 
to be interpreted here as a symbol for any quantity 
other than the ratio defined) and that it is de- 
sired that k = k„„ its maximum value consistent 
with equation (3), above. Let it also be assumed 
that R is approximately equal to R,n., the maximum 
permissible value of R,, that Cg = 2Cx, and that 
C„ = C„„„ the maximum permissible shunt capaci- 
tance specified for the crystal unit. Then, 


(max) Rg = k.„ (min) PI = Re,„ 

also 


(max) Rg = >/2irr / 2u C^V R,,., 


Thus, 

(max) Rg/(min) PI = k^ = 


toC, •%/ 2^ r R 

em 

2^/ &) Com 


1—300 (4) 
or 

km“ = ■»■«• 1 (min) Xc„ | /2 (min) PI 

1—300 (5) 


It will be found in practice that the limiting values 
of km given by equations (4) and (5) are normally 
smaller than the minimum desired value of 15. If 
this should be the case in an actual circuit, the 
assumption that the power losses in the grid cir- 
cuit are negligible can no longer be made, and the 
actual value of k^ would be even less than that 
given above. The factor \/R/fE in equation ( 1 ) is 
derived upon the assumption that only the crystal 
losses are significant. If this is not to be the case, 
this factor should be replaced by one equal to 
/ total energy expended per cycle 
V energy stored ‘ ’ 

though it would seem from equations (4) and (5) 
that Rg cannot be safely made more than 5 to 10 
times larger than the minimum PI, particularly if 
an AT cut is employed, since it has a value of r of 
only 250, and since Cg is normally no greater than 
4 or 5 times Cm„, it should be remembered that the 
value of km above is based upon the assumption 
that no energy is being fed to the circuit during 
an entire cycle, so that the net loss is equal to the 
gross loss. This condition is only approached in 
high-efficiency class-C circuits where the operat- 
ing bias is several times the cutoff bias. Except 
in the case of power oscillators, such operation is 
not feasible because of the high operating voltages 
that are required. The larger the fraction of the 
cycle during which the tube is conducting, the 
larger the ratio of the usable Rg to that given by 
equation (3) . If the tube is conducting one-half the 
time, class-B operation, the maximum safe R, is 
more than twice that given by equation (3). For 
class-B and class-C operation, the output stability 
is almost entirely dependent upon the automatic 
adjustment of the bias, for any decrease in signal 
strength will mean a decrease in over-all amplifi- 
cation unless the bias can drop immediately to 
allow more energy to be fed to the circuit. On the 
other hand, it was found in paragraph 1-294 that 
if an oscillator tube is operated at a bias imme- 
diately above the knee of its E.-L curve, the bias 
can remain fixed and the variations in excitation 
directly produce a change in amplification that 
tends to annul the original variation. If the tube 


WADC TR 56-156 


146 



voltages are so selected that a gridleak bias at 
equilibrium is also at the optimum fixed-bias point, 
then limiting can be achieved both from the grid- 
leak action and the excitation swings. Under these 
conditions, R* can be safely increased to values 
beyond one megohm, even at high frequencies. As 
a design consideration, however, the gap between 
the theoretical and the practical solution can prove 
quite wide. Among the optimum-bias bugs that 
resist extermination; there is the difficulty of find- 
ing a vacuum tube having the desired operating 
characteristics, and once found, there is the addi- 
tional problem of maintaining an optimum oper- 
ating state with crystal units having different 
values of effective resistance. These problems are 
discussed in some detail in succeeding paragraphs. 
The main problem is to reduce the grid losses to 
negligible proportions without endangering the 
output voltage stability. This can normally be done 
with any parallel-resonant crystal oscillator if the 
tube is conducting throughout most of each cycle. 

Gridleak Resistance and Output Control 
in Pierce Circuit 

1-301. If it is necessary for a Pierce oscillator to 
provide a higher voltage output than can be ob- 
tained under the conditions of maximum fre- 
quency stability, the C^/C„ ratio can be increased. 
If the total load capacitance is to remain constant, 
Cg will necessarily be larger, and the excitation 
voltage smaller, so a smaller value of R, can be 
used without the grid losses becoming significant. 
If the capacitance ratio is to be adjustable in order 
to permit an operator or technician to control the 
output voltage, R^ cannot he made larger than that 
value which would permit a stable output with the 
largest operable value of Cg at the highest fre- 
quency at which the oscillator is to be used. If such 
an adjustment is to be provided in a Pierce circuit, 
Cg and C„ should be so ganged as to always provide 
a constant load capacitance. This problem is dis- 
cussed in paragraph 1-318. Insofar as the grid-to- 
cathode resistance is concerned, the maximum 
safe value of R, becomes less if Cg is to be variable 
than otherwise. Without changing the Cg/C„ ratio, 
larger outputs can be achieved by reducing the 
value of Rg to a point where the grid leakage is so 
great that the average bias is considerably smaller 
than the peak excitation voltage. With this the 
case, the oscillations must build up to higher amp- 
litude levels before equilibrium can be reached. Al- 
though the maximum excitation is still fixed by the 
rated drive level of the crystal unit, the output can 
be controlled somewhat within this restriction by 
a variable R,. At the higher frequencies, this 


Section I 
Crystal Oscillators 

method of output adjustment requires such low 
values of R* that the grid losses seriously affect 
the frequency stability. However, at very low fre- 
quencies, a variable R^ could be feasible as a means 
of adjusting the output of a Pierce circuit to a 
desired level when one crystal unit is replaced by 
another of different effective resistance. Although 
such a design feature has no particular recom- 
mendation, it could be preferred over those meth- 
ods of output control that require adjustment of 
the Cg/Cp ratio, which risk changes being made in 
the total load capacitance. With grid control, the 
lowest adjusted value of R^ could be designed to 
provide the desired output when a crystal unit of 
maximum effective resistance (minimum PI) is 
connected in the circuit; whereas the larger values 
of R, could ensure the same output with some 
theoretical minimum value of effective resistance. 
Since the crystal current, U, is practically constant 
as long as the output voltage E,, is constant, the 
power losses in the crystal, equal to U-'R,., tend to 
vary directly with R„ as long as E„ is held constant 
by adjustments of Rp. Under those conditions 
where the capacitance ratio does not change, a 
maximum crystal drive level is required for the 
crystal unit of maximum R,., and a minimum crys- 
tal drive level when R,. is a minimum — the reverse 
of those conditions discussed in paragraph 1-294 
when a fixed bias instead of a fixed output is 
assumed. 

1-302. As applied to crystal oscillators in general 
it cannot be said that a variable gridh'ak resistance 
is advisable except for test purposes or unless its 
purpose is to obtain the minimum possible grid 
losses when changing from one crystal unit to 
another. As an output-voltage control device other 
methods are generally to be preferred. Except at 
very low frequencies, the resistance values neces- 
sary to appreciably lower the average bias are too 
small to prevent the grid losses from becoming a 
significant frequency-determining factor. This 
statement, of course, only expresses a general rule, 
and in specific instances the inter-relations among 
the circuit variables may be such as to annul the 
effects upon the frequency. For example, the bot- 
tom set of curves in figure 1-129 (A) is to be ex- 
pected theoretically to indicate a greater frequency 
stability when R^ is 1 megohm rather than when 
it is 0.2 megohm, but this effect was not observed. 
Figure 1-130 shows curves of output voltage ob- 
tained from the same experimental oscillator that 
was used in plotting the curves of figure 1-129. 
Although the curves are plotted as output-voltage 
versus crystal driving power, it should be under- 
stood that the actual independent variable for each 


WADC TR 56-156 


147 



Section I 

Crystal Oscillators 

curve is the plate voltage. Each curve represents 
a particular value of grid resistance. The cross 
lines intersect the curves at points corresponding 
to the same values of plate voltage. From figure 
1-130 it can be seen that large percentage changes 
in the grid resistance can cause changes in the 
output voltage on the order of 30 per cent or so, 
but which increase sharply as becomes small. 
1-303. If an ad j usable output voltage is desired, 
probably the best solution to the problem is to use 
a screen-grid tube having an r-f-bypassed, varia- 
ble, voltage-dropping resistor in series with the 
screen supply voltage. Varying this resistance will 
control the output of the tube and the crystal driv- 
ing power. The maximum permissible output volt- 
age must be determined on the assumption that 
the crystal unit has the maximum permissible R,. 
E 

Since Ig «= y — maximum permissible 

Ig = V^cm/Rem, wherc P,.m and Rem are the maxi- 
mum crystal driving power and effective resist- 
ance, respectively, then the maximum permissible 
constant Ep is 


(max) Ep = (X, -f Xc,) V Pom/R,™ 

1—303 (1) 


X 

If it is assumed that Xc, Xop » , where 

X,(= — l/mCi) is the total load reactance equal 
and opposite to X,, equation (1) becomes 


(max) Ep = 



2a>C, 


1—303 (2) 


According to equation (2), the maximum permissi- 
ble constant Ep varies inversely with the crystal 
frequency. If the oscillator is to be used at more 
than one frequency, and at the same time is to 
provide the same output voltage regardless of the 
frequency, the maximum Ep is that value given by 
equation (2) for the crystal unit of highest fre- 
quency, assuming the crystal specifications are the 
same for all frequencies. With the ratio of Cg/Cp 
approximately equal to one, equation (2) also gives 
the value of Eg, which obviously will also remain 
constant. R, can be made quite large, so that |Ec| 
will approximately equal E„\/2^. With Ec constant, 
and with the plate voltage Ep also assumed to be 
constant, the operating position of the tube on the 
Eolp curve largely becomes the function of the 
screen voltage. As the screen voltage is increased, 
g„, increases, which means that the slope of the 



CRYSTAL DRIVE- MILLIWATTS 


figure I-I30. Output curves of tuned Pierce oscillator for different values of grid resistance when reactance of 
tuned plate circuit is adjusted for output voltages equal to 50 percent of the maximum attainable. A 7-ific 
CR-I8/U crystal unit was used, having a PI of 49,000 ohms when operating into its rated load 

capacitance of 32 /i/J 


WADC TR 56-156 


148 



$«cHon I 
Crystal Oscillators 


EJb curve becomes steeper. Also, the cutoff bias 
is increased. Since E, is being held constant, the 
effect is one of shifting the operating bias, percent- 
age-wise, closer to or farther up the straight por- 
tion of the EJi, curve. From the point of view of 
using as large a value of Kg as is possible, it is 
desirable that the operating position be just above 
the knee of the EJb curve when the screen voltage 
is to be a maximum, i.e., when R, of the highest- 
frequency crystal unit is a maximum. 

Gridleak Resistance and Oscillator Keying 
of Pierce Circuit 

1-304. If avoidable, a crystal oscillator should not, 
itself, be keyed. For one reason, the oscillation 
build-up time is not negligible if rapid telegraph 
keying is desired. As the operable speed limit is 
approached the wave shape becomes distorted and 
the harmonic output is considerably increased. 
Even the keying of a crystal oscillator in a push- 
to-talk voice transmitter is not desirable if fre- 
quency stability is important, since on-and-off 
operation constantly raises and lowers the crystal 
temperature. Thus, the frequency is kept in a state 
of constant variation to a degree dependent upon 
the magnitude of frequency-temperature coeffi- 


cient of the crystal unit at the average operating 
temperature. Unless necessary for reasons of econ- 
omy in space, cost, or the like, the oscillator should 
be designed for continuous operation and the key- 
ing performed in one or more of the succeeding 
amplifier stages. Usually the keying circuit is de- 
signed to remove and apply by one means or 
another, a cutoff bias in the buffer-amplifier stage. 
During the time that the buffer amplifier is cut off, 
the crystal circuit continues to oscillate, but the 
signal cannot be amplified and applied to the suc- 
ceeding stages. 

1-305. When it is necessary to key the oscillator, 
itself, the reason is normally that the space and 
weight requirements are so limited that no more 
than one or two vacuum-tube stages can be al- 
lowed. For this same reason, the oscillator is prob- 
ably required to develop as much output power as 
possible, so that a Miller, rather than a Pierce cir- 
cuit is generally employed if crystal control of the 
frequency is required. Nevertheless, the factors 
affecting the build-up time are approximately the 
same in either circuit. Fundamentally, the reason 
that a crystal oscillator requires a relatively much 
longer build-up time than does a conventional in- 
ductor-capacitor tuned circuit of the same reso- 



WAOC TR 56-156 


149 


Section I 

Crystal Oscillators 

nant frequency, is because the energy to be stored 
in the crystal is much greater than that which 
would be stored in an inductor-capacitor circuit. 
For a given tank current, the stored energy 
is proportional to the inductance, so, to a first 
approximation, we can suppose that the build-up 
time of, say, a Pierce oscillator as compared with 
that of a Colpitts oscillator of the same frequency, 
is directly proportional to the inductance ratio. On 
the other hand, it can be imagined that the 
build-up time tends to vary inversely with the total 
effective resistance in the tank circuit. The greater 
this resistance, the more quickly do the losses in 
the circuit rise to equilibrium with the rate of en- 
ergy supply. The build-up time also tends to vary 
inversely with the frequency. Clearly, if the fre- 
quency were one cycle per second, equilibrium 
could not be reached in a shorter period. Finally, 
the build-up time is a function of the electro- 
mechanical coupling of the crystal to the circuit. 
The larger the C„/C ratio of the crystal unit, the 
weaker is the coupling and the longer is the period 
before equilibrium can be reached. The exact rela- 
tions of all the circuit variables in an equation 
expressing the time required for the amplitude to 
rise to within one per cent or so of its equilibrium 
limit would, indeed, be quite involved. Insofar as 
the crystal is concerned, the build-up time can be 
expected to vary positively if plotted against L, C, 
Co, and 1/R of the crystal unit. The percentage 
variation of the build-up time with a given per- 
centage variation in L can be expected to be 
greater than with the same percentage variation 
in C, because of the fact that, say, an increase in C, 
although increasing the build-up time by lowering 
the frequency, will also tend to decrease the build- 
up time by improving the electromechanical ratio. 
Thus, if the frequency remains constant, a crystal 
oscillator can be keyed at a faster rate if the L/C 
ratio is kept to a minimum, provided C„ is not in- 
creased. In other words, a crystal element should 
be chosen that has as large a piezoelectric effect as 
possible, provided the frequency-temperature co- 
efficient is small. For example, for high-frequency 
circuits, an AT-cut crystal which has a capacitance 

(] 

ratio 250 is to be expected to provide better 

keying characteristics than a BT-cut crystal, 
which has capacitance ratio of 650. Preferably, 
from the point of view of a maximum keying speed 
for a given output voltage, the gridleak resistance 
should be kept small, not only to load the circuit 
and to provide quick-action limiting, but also to 
keep the positive swings of the grid and the trans- 
conductance high. The oscillator will almost cer- 


tainly be designed for maximum power output, so 
that the tank circuit will be well loaded, for which 
reason the grid resistance must be kept relatively 
small as a safeguard against intermittent oscilla- 
tions. It is questionable as to just how much the 
effective tank resistance limits the build-up time. 
Of course, if the resistance were zero, the oscilla- 
tions would theoretically continue to rise indefi- 
nitely. On the other hand, the time required for 
the amplitude to reach any given value is least 
when the energy being lost from the circuit is 
least. In this respect, the build-up time tends to 
vary directly, not inversely, with the tank resist- 
ance. It would seem, that to obtain a maximum 
keying speed it might be preferable to use a fixed 
bias or a cathode bias, instead of the gridleak ac- 
tion. Using a sharp-cutoff tube biased for class-A 
operation, a grid, plate, or output circuit limiting 
arrangement could permit the oscillations to build 
up to a given level under conditions of a maximum 
ratio of input to dissipated power. Above this amp- 
litude level the ratio would sharply decrease. Such 
a circuit could raise the permissible keying speed, 
but since this is accomplished by virtue of sudden 
changes in the circuit parameters, which changes 
always accompany to some extent any limiting ac- 
tion, an increased frequency instability and har- 
monic output are almost certain to result. Although 
a crystal oscillator should not be designed to be 
keyed unless absolutely necessary, experimental 
circuits have obtained keying speeds approaching 
400 words per minute. The higher the keying 
speed, however, the greater must be the frequency 
tolerance. 

Gridleak Resistance When Pierce Circuit Permits 
Switching from Crystal to Variable LC Control 

1-306. It is often necessary to provide a variable- 
tuned, inductor-capacitor auxiliary circuit to per- 
mit emergency operation at frequencies other than 
those provided by the available crystals, or in the 
event of crystal failure. For this purpose it is often 
possible and is usually desirable to use the same 
vacuum tube that is used during crystal control. 
For example, a Pierce circuit could be readily con- 
verted to a Colpitts circuit simply by switching 
from the crystal to a tuning inductor, or to an 
inductor shunted by a variable capacitor. However, 
when such a conversion is made, the ratio of the 
stored energy to the power dissipation becomes 
much smaller than that during crystal control. For 
this reason, the maximum safe value of gridleak 
resistance is much smaller than during crystal 
operation. For output voltages comparable to those 
obtained with crystal control, the LC circuit em- 


WADC TR 56-156 


150 



ploys gridleak resistances ranging ^011/20, 000 to 
occasionally 100,000 ohms. If the LC ydrcuit is in- 
tended to furnish a much greater output than the 
crystal circuit, lower values of lymay be neces- 
sary. Rather than require the^ystal circuit to 
operate with small values of R*y^t would be prefer- 
able to connect an additional ainunt resistor in the 
grid circuit when switching to variable-tuning 
control. 

Gridleak Resistance When Used with Cathode 
Biasing Resistor in Pierce Circuit 

1-307. In addition to the voltage across the grid- 
leak resistance, part of the bias voltage can be 
furnished by an r-f-bypassed resistance in the 
cathode circuit. The cathode resistor protects the 
tube from excessive plate current should oscilla- 
tions cease, and has the additional advantage of 
reducing the grid current and, hence, the grid 
losses. The power expended in the grid circuit will 
be approximately equal to where Et is the 
total bias and I,, is the grid current. Actually, un- 
less an r-f choke is used in the grid circuit, the grid 
losses will be somewhat greater than E^Ic because 
of the a-c component of voltage across Rj. As the 
cathode component of the bias becomes small, the 
grid losses approach 1.5 as a limit. See para- 
graph 1-296. The values of the cathode resistance, 
Rk, usually range from 100 to 1000 ohms. The re- 
actance of the bypass capacitor should be at least 
as small as Rk/10 at the lowest operating fre- 
quency. With Rk connected between cathode 
and ground, the d-c voltage developed equals 
( Ib -f- 1. ) Rk ; or approximately, ERk. The total bias. 


Saction I 
Crystal Oscillators 

E„ is still approximately equal to V2 E,. The d-c 
grid current is given by the equation 

I E, I - Ek V2 E, - E Rk 

R. R. 

1—307 (1) 

where Ek is the voltage across the cathode resistor. 

AGC USED WITH PIERCE OSCILLATOR 

1-308. Where space and cost permit, optimum out- 
put stability can be had when the oscillator bias 
is provided through an automatic-gain-control cir- 
cuit. Gridleak action can be effective in initiating 
oscillations, but the bias furnished through AGC 
should be of much greater magnitude in order to 
be of maximum effectiveness. A small increase in 
output voltage must cause a large increase in bias. 
The use of AGC reduces the grid losses to a mini- 
mum and maintains a constant amplitude of os- 
cillation. It is this latter feature that is, of course, 
of most importance — particularly so when the 
same oscillator is to be switched from one crystal 
unit to another. The voltage requirements for con- 
stant output without risking the overdrive' of any 
of the crystals are the same as those that apply in 
the case of manual adjustment of the output. (See 
paragraph 1-302.) An A-G-C circuit applicable for 
use with a Pierce, or Miller, type oscillator, is 
shown in figure 1-131. The oscillator output is amp- 
lified by Vj. The output of V2 is then rectified by 
V:,. The oscillator bias equals the average rectified 
voltage across R3. C, bypasses the r-f component 
to ground. If R, were increased indefinitely the 



WADC TR 56-156 


Figure I-I3T. Pierce osciilator with automatic gain control 

151 




Section I 

Crystal Oscillators 

bias voltage would approach in magnitude the peak 
value of the Vj output voltage. Ri, Rj, and R, 
not critical — each can be made equal to 50K/U a 
faster-acting gain control is required. However, 
R, should be kept as large as possible. Ass^e that 
the r-f losses in R,, R,, and Rj are negl^ble and 
that C,, and Cg are approximately equal, 4o that V, 
is operating into a load impedance ai^roximately 
equal to PI/4. Under these conditions gm will be 
the minimum and Rp the maximum possible for 
sustained oscillations as long as the load capaci- 
tance across the crystal unit remains constant. The 
actual values of g„, and R„ are fixed by the vacuum- 
tube characteristics. Although the effective pa- 
rameters of the tube are directly dependent upon 
the peak-to-peak magnitude of the excitation, as 
well as indirectly through the bias, it can be said 
that to a first approximation the equilibrium Rp 
and g,„ are associated with a bias of more or less 
definite magnitude if the plate voltage is constant, 
and that approximately the same bias must exist 
regardless of whether it is developed by gridleak 
action or by AGC. Thus, the difference between 
AGC and gridleak control is not primarily in the 
magnitude of the bias, but in the amplitude of os- 
cillations. Gridleak action requires that the peak 
excitation voltage of Vj be slightly greater than 
the required bias ; AGC requires that the peak ex- 
citation voltage of V 2 times the voltage amplifica- 
tion of the V 2 stage be slightly greater than the 
required bias of V,. If the peak excitation voltage 
of Vj is assumed to equal Ep^, which, in turn, is 

assumed to equal 1^, and k 2 is the effec- 

tive amplification of tne V, stage, then 

I E, I = kj E,™ 1-308 (1) 

or 

Egm = I Ec/kj I 

Since E,. is approximately fixed, it can be seen that 

the amplitude of oscillations is only ^ as large as 

those that would exist by tlie gridleak method em- 
ploying the same plate voltage. This is not a de- 
sirable feature where large output is required, but 
from the point of view of ensuring a low crystal 
drive and maximum stability, an A-G-C circuit has 
great advantages. Although the equilibrium amp- 
litude is low, oscillations start as readily as with 
gridleak bias. AGC permits class-A operation with 
remote-cutoff tubes, and, since the limiting is very 
slow-acting, very pure sine-wave outputs and ex- 
cellent frequency stability as well as amplitude 
stability is obtainable. 


PLATE-SUPPLY CIRCUIT OF 
PIERCE OSCILLATOR 

1-309. For optimum frequency stability it is im- 
portant that the r-f impedance of the B+ circuit 
be as high as possible relative to the impedance 
of the tank. If Cg/Cp = 1, the tank impedance 
equals PI/4. If the oscillator is intended to oscillate 
at only one frequency, or within a narrow fre- 
quency, range, it is generally preferable that the 
B+ voltage be fed through an r-f choke. This 
method affords a maximum impedance with mini- 
mum loss and minimum voltage at the B+ source. 
The inconvenience of an r-f choke is that its im- 
pedance changes with frequency, being inductive 
below its effective parallel-resonant point, and ca- 
pacitive above. As long as this effect does not 
change the effective value of C„ by more than ±10 
per cent, the total load capacitance will not change 
by more than 5 per cent, if Cg/Cp = 1. Within these 
limits the use of a choke is to be preferred. For 
wide frequency ranges, a resistor should be used 
in the plate circuit, such as R, in figure 1-131. It 
is desirable for this resistance to be as high as 60K, 
or higher, from the point of view of frequency 
stability. On the other hand, the larger the resist- 
ance the higher the B+ voltage source must be to 
provide a given plate voltage. Plate-supply resist- 
ances on the order of 5000 to 10,000 ohms have 
one other important advantage besides permitting 
lower B+ sources. They load the oscillator tank so 
that differences in the resistance of the crystal 
from one unit to the next have very little effect 
upon the output impedance of the tube. Hence, . 
when a change is made from one crystal unit to the 
next, the output voltage remains approximately 
the same. 

1-310. The proper compromise in selecting a plate- 
circuit resistance depends upon the frequency-tol- 
erance limits. The plate-circuit resistance does 
afford a certain frequency-stabilizing effect that is 
not provided by an r-f choke, particularly so when 
age is used. The effect is one of reducing the 
change in Rp of the vacuum tube caused by a 
change in grid bias. For example, if the bias be- 
comes more negative Rp increases, and Ib, the aver- 
age plate current, decreases. There is then less 
voltage drop across the plate-supply resistor, and 
the resulting increase in plate voltage tends to de- 
crease Rp, thereby annulling part of the increase 
in Rp due to the change in bias. The plate-voltage 
source should be regulated, if good stability is re- 
quired. Where the frequency deviation must be 
kept to a minimum, the oscillator may require a 
separate rectifier unit, filter circuit, and voltage- 
regulator circuit. 


WADC TR 56-T56 


152 



Saction I 
Crystal Oscillators 


CHOOSING A VACUUM TUBE FOB 
THE PIERCE CIRCUIT 

1-311. It is no problem to find a vacuum tube that 
will permit a Pierce circuit to oscillate. Indeed, one 
of the major problems in tube circuit design is to 
prevent oscillations from occurring. With a crystal 
connected between the plate and grid of any vac- 
uum-tube amplifier, the stray capacitance in the 
circuit is usually sufficient to cause oscillations to 
build up. If the plate voltage is not so high that 
the crystal is over-driven, the frequency stability 
of a stray-capacitance circuit may even be satis- 
factory for general-purpose use. Thus, the problem 
is not to find a vacuum tube that will work, but 
one that will be most satisfactory from the point 
of view of output stability and cost. First, a large 
tube is not necessary, since the Pierce circuit is 
not suited for large output. The choice of tube will 
depend somewhat upon the exact purpose of the 
oscillator and of the equipment of which it is a 
component. If the frequency tolerance is to be 
large, little thought need be given to fine points in 
the design, for the principal problem will be to 
keep the production costs to a minimum. A triode 
would be satisfactory, a 5K to 50K resistance in 
the plate circuit, a Cg/Cp ratio between 1 and 2, 
and a plate voltage sufficiently low so that the driv- 
ing power of the crystal does not exceed the rated 
level for any effective crystal resistance meeting 
the specifications. A high-mu triode generally pro- 
vides the better frequency stability because of its 
larger effective Rp, but it will have a higher plate 
dissipation for the same output voltage. Of the 
high-mu triodes, probably the 6AB4 is to be pre- 
ferred as a simple unit, and the 12AX7 and the 
12AT7 as twin triodes. It is the medium-mu tube 
that has been the most favored by design engi- 
neers when a triode has been chosen. Of these the 
6C4, 6J4, and 7A4 single units, and the 6SN7-GTA 
twin unit are among the more popular. The 6C4 
and the 6J4 are to be preferred for high-frequency 
operation. Since the 7A4 and the 6SN7-GTA have 
approximately 4 iijd capacitance between grid and 
plate, the 6C4, 6J4, 6.16, or the 12AU7, each with 
1.5 nnf capacitance grid to plate, should provide 
the better frequency stability — particularly at 
high frequencies. The 7A4 and the 6SN7-GTA are 
generally more satisfactory for use in a Miller cir- 
cuit. Where greater frequency stability is required, 
a pentode should be used. A pentode has the ad- 
vantages of low plate-to-grid capacitance, greater 
Rp, and a screen grid whose voltage can be ad- 
justed independently of the control-grid bias and 
plate voltage, thereby permitting a greater range 
of adjustments in the plate characteristics. Con- 


ventional pentodes must be operated at reduced 
voltages, to avoid overdriving the crystal, unless 
rather high Cg/Cp ratios are used. Subminiature 
pentodes have operating characteristics at their 
normal operating voltages ideally suited for crys- 
tal drive levels. The 1U4 is one such type having 
a sharp cutoff. Among the miniature pentodes hav- 
ing a sharp cutoff, the 6AU6, 6BC5, and 6AH6 are 
tubes generally recommended for wide-band, h-f 
circuits. The 6CB6, although designed principally 
for television use at 40 me, should also be quite 
appropriate in crystal oscillator circuits. Remote- 
cutoff tubes are generally used only in special cir- 
cuits. For example, if a low harmonic output is 
required, such tubes could be employed in conjunc- 
tion with AGC. When the harmonic content is not 
of first importance, AGC is more effective if used 
with sharp-cutoff tubes, where a slight change in 
grid bias can make a much larger change in small- 
signal outputs than is possible if the slope of the 
E..I|, curve changes very gradually. Actually, re- 
mote-cutoff tubes, when used, are usually found in 
doubler circuits, because of the large second-har- 
monic component that is produced. Although 
class-B and class-C operation with sharp-cutoff 
tubes can produce even greater harmonic outputs, 
there is the problem of ensuring that a crystal of 
large R,. will not be overdriven if it is to be oper- 
ated in a class-B or class-C circuit. The output 
voltages of remote-cutoff tubes tend to vary more 
with crystals of different resistances than is the 
case when sharp-cutoff tubes are used. The reason 
is that in the former case the effective I,, continues 
to increase as R, becomes small, since very large 
excitation voltages are required to override the 
cutoff point. On the other hand, the effective Ip 
begins to decrease when class-B operation is ap- 
proached and such operation can be had with rela- 
tively small excitation voltages when sharp-cutoff 
tubes are used. If a remote-cutoff tube is desired, 
recommended types are the subminiature 1T4, the 
miniature 6BA6 and 12BA6, the lock-in 7A7, and 
the conventional-sized tubes such as the 6SK7 and 
12SK7. The mention of particular vacuum tubes 
here should not be construed as official recom- 
mendation; they are named simply because they 
are the tubes commonly found in new equipment. 
The design engineer may very well find that the 
characteristics of other tubes are more appropri- 
ate for his needs. 

Pierce-0 scillator Design Considerations When 
Vacuum Tube with Very Sharp Cutoff is Used 

1-312. In making a preliminary approximation as 
to what the performance of a particular tube will 


WADC TR 56-156 


153 


Section I 

Crystal Oscillators 


be if used in a Pierce circuit, it should first be 
kept in mind that the ratio 


”■ Cg/Cp 


mZl 

Rp + Zl 


gm Rp Zx, 

Rp + Zl 
1—312 (1) 


is the gain of the tube. If the gain = k, and if 

R 

— 2 . is 10 or greater, then 
Zl 


gm 


k/Z,. 


1—312 (2) 


or 

Rp = MZL/k 1—312 (3) 

Either equation (2) or (3) can be used to esti- 
mate approximately the grid bias for a given plate 
voltage, and vice versa, that can be expected if a 
particular tube is used. Assume, for example, that 
k = 1 , that gridleak bias is to be used, and that 
the grid and load losses are negligible compared 
with the crystal driving power. In this case, the 
minimum expected Zl will equal (min) PI/4, 
which occurs when a crystal unit has the maximum 
allowable and is operated at the rated load ca- 
pacitance, Cj. Under these conditions, the maxi- 
mum permissible bias, as given by equation 1-293 
( 2 ), is 

(max) Ee = - V 2 Pen, / 2a)C, V R,„, 

This maximum value of E,. is to be interpreted as 
a maximum that can be allowed only if R„ is a 
maximum or if the output voltage is to be the same 
magnitude regardless of the value of R,. In this 
latter case, P,.„, and R,.„, fix the output and bias 
limits for all crystal units of a given type. The 
constant output can be obtained in several ways: 
by the use of an actual or equivalent, parallel, plate 
load resistance that is small compared with the 
minimum nonloaded crystal tank impedance; by 
the use of AGC, by the use of manual voltage ad- 
j ustments ; or by other methods. The present dis- 
cussion concerns only the noncontrolled nonloaded 
circuit. If a crystal unit of maximum R„ being 
driven at the maximum drive level, is replaced by a 
crystal unit of smaller R„ Zl increases, and Ep and 
Ig tend to increase proportionately, so that the crys- 
tal driving power, equal to 1 *= Re, is greater than 
when Re is a maximum. According to equation (3) , 
insofar as it can be assumed that /* remains ap- 
proximately constant (in practice, decreases 
somewhat) R,. increases proportionately with Zl, 
so that although the equivalent generator voltage, 
—y-Eg, increases, I,, remains constant. Thus, Ip => 
g,„Eg kEg/ZL constant. In an actual circuit 


where Rp > > Zl and the vacuum tube has a very 
sharp cutoff, the effective Ip increases up to the 
point that the tube is cutoff for approximately 
three-fifths of the negative alternation (three- 
tenths of the entire cycle) . As the excitation volt- 
age increases beyond that point. Ip progressively 
decreases, although the total power supplied to the 
tank circuit continues to increase as long as the 
excitation voltage continues to increase. The con- 
clusions above are derived in the special case of a 
Cg/Cp ratio of unity, by assuming that for all prac- 
tical purposes the plate-current pulses are in phase 
with Ep, and that Ep > > Ep. Figure 1-132 illus- 
trates different states of operation of the same 
oscillator circuit that can occur if crystal units of 
the same frequency but different values of Rp are 
inserted in the circuit. A change from the class-A 
to the class-C state could readily occur if the crys- 
tal Rp were reduced by more than one-half. The 
effective Ipm is defined by the equation 

P 7 .L = Ipn. Ep,„/2 1-312 (4) 

where Pz,. is the power expended in the tank cir- 
cuit. Since Z,, is very small compared with Rp, it 
can be assumed that the sinusoidal component, Ep, 
of the with-signal d-c plate voltage, ei„ is negligi- 
ble by comparison with the average value, Ep ; that 
is, Ep ±: Ep„, «= Ep. With this assumption we can 
treat Ip„„ the value of the with-signal, d-c plate 
current, at the positive peak of excitation (e,. = 0 ) 
as a constant. The assumptions above also imply 
that very little grid current exists ; otherwise, the 
larger excitation voltages would drive the grid 
considerably above zero at the positive peaks. With 
the peak instantaneous d-c plate current a con- 
stant, the total energy supplied by the power 
source progressively decreases as Zl and the ex- 
citation increase, since Ip, the average ip, becomes 
progressively smaller, whereas Ep remains con- 
stant. (Actually, if the plate current is supplied 
through a resistor, a decrease in Ip causes Ep to 
increase somewhat. For the problem at hand, 
assume that a regulated B+ is fed through an r-f 
choke.) Thus, it can be seen that as Rp becomes 
small the plate efficiency increases considerably. 
However, the efficiency of a crystal oscillator does 
not approach the high ratios of input to tank 
power that are obtained with conventional class-C 
power amplifiers and oscillators. The latter cir- 
cuits can operate at efficiencies of 60 to 90 per cent 
because approaches Ep in magnitude. The in- 
stantaneous power being dissipated in the tube is 
the instantaneous value of ipOp, and the instanta- 
neous power being delivered to the tank is ipep. 
When ip is a maximum, ep = Ep — Epm < < e^ = 


WADC TR 56-156 


154 



S«cHon I 
Crystal Oscillators 


E,kb, so that most of power goes to the tank circuit. 
In the conventional Pierce oscillator such high 
efficiency is not to be approached unless the C*/Cp 
ratio is to be made very large and Eb approaches 
in magnitude the voltage specifications of the crys- 
tal unit. Now, to obtain a maximum output with- 
out the risk of overdriving a randomly selected 
crystal unit, it will be useful to derive approxi- 
mate equations concerning the change in crystal 
power with a change in R*. The crystal power, we 
shall assume to equal the total tank power, Pzl. 
In short, the problem is to be able to express Pz, 
as a function of R,. Ibm, Et, and E™ (the cutoff 
voltage) will be considered constants, and ip and ep 
are to be assumed to be in phase. First, we express 
the effective I,„„ for each class of operation in 
terms of the constants above and the angles <t) and 
6, where appropriate. (See figure 1-132.) As a 
safeguard against intermittent oscillations, which 
are most likely to occur when R^ is a maximum, 
assume that the bias for maximum R^ is to occur 
on the straight portion of the Eelb curve. If the 


oscillations are to build up at all, they must con- 
tinue to do so until the negative excitation peak 
at least extends into the lower bend of the EcIb 
curve, for it is only beyond the straight portion 
of the curve that g„ can change in order to seek 
its equilibrium value — that is, unless R* is so small 
that equilibrium is reached by virtue of the in- 
crease in grid losses alone. With a large Rg and 
a reasonably sharp cutoff, it is virtually impossible 
for oscillations to start if the amplification is not 
at least sufficient to increase the excitation to 
where the negative peak is very nearly equal to 
E„. Assume, then, that with R, — Rem, the oscil- 
lator is designed to operate approximately as 
shown in figure 1-132 (A). It can be seen intui- 
tively that 

(Class A) Ipm - Ibn./2 1-312 (5) 

and with C,/C„ = 1, considering only the unsigned 
magnitudes of the bias voltage, 

(Class A) Epm = Ee = Ee„/2 1-312 (6) 


LARGE Rg — — K SMALL Rg 



Fi'Sure I-I32. Change of state of Pierce oscillator with a Cp/Cp ratio of one, under no-load conditions when 
Cl, is held constant and the effective resistance of the crystal changes 


WADC TR 56-156 


155 


section I 

Crystal Oscillators 

so that 


(Class A) Pzl = Ip„ Ep„/2 = Ibm Eeo /8 

1—312 (7) 


Ecolbn, /*' + 

2T(l + 8in«)y^ 


♦ 

(sin^ wt + sin 4> sin o>t) doit 
1—812 (16) 


Equation (7) represents the maximum possible 
crystal power if a tube is not to be driven beyond 
cutoff. Referring now to figure 1-132 (B), we shall 
assume that a crystal unit with an R. slightly less 
than the maximum is connected so that the bias 
is similar to that under AB operating conditions. 
In, represents the apparent maximum Ip. It can be 
seen that except for the angle (w — 2 </>), when 
the tube is cut off, 

ib = In, (sin cot -|- sin 0 ) 1 — 312 ( 8 ) 

where <t> can be considered a constant. Now, 

Ibm = Im (1 + sin 0 ) 1 — 312 (9) 

so, on substitution in equation ( 8 ), 

ib = , , (sin cot -I- sin 0 ) 1—312 (10) 

1 "T Sin <t> 

Similarly, 

Eco = Eg,n (1 -f sin 0 ) = Ep„, (1 - 1 - sin 0 ) 

1—312 (11) 

so that 


On integration. 


(Class AB) Pzl = 


Eco Ibm (x + 2 0 + 2 sin 2 0 ) 
4t (1 -|- sin 0 )* 


1—312 (16) 


No maximum exists for equation (16) with values 
of 0 between 0 and ir/2. For class-A operation 
similar to that in figure 1-132 (A), 0 = ir/2, so 
that equation (16) becomes 


(Class A) Pzp = E.. Ibm /8 

This checks, as is to be expected, with equation 
(7) . For class-B operation, 0 = 0, so that equation 
(16) becomes 


(Class B) Pz. = E,„ Ibm/4 1—312 (17) 

Note that the power expenditure in the crystal 
unit for class-B operation is exactly twice that 
found for class-A operation. Since Epm under 
class-B conditions is equal to Ec„ (see figure 1-132 
(C)), or twice the class-A value of Epm, then, be- 
cause Pzt = IpmEpm/2, the effective Ipm at class B 
must be equal to the same effective value as at 
class A. Thus, 


Cp = Epm sin cut 


E 


CO 


1 -t- sin 0 


(sin cut) 


1—312 (12) 


Since no energy is being supplied during the time 
that the tube is cut off, the energy delivered to 
the tank per cycle is 



^ (»■ + «) 
Cp ib dt 

(- 0 ) 


if- 


IT + 

Cp ib dcut 


1—312 (13) 


where t = time in seconds. 

Thus, the energy delivered per second, is 


P 


Zt 



ib dcut 


I 


Cp ib dcut 


1—312 (14) 


On substitution of Ep and Ib from equations (10) 
and ( 12 ), 

WAOC TR 56-156 


(Class B) Ipm = Ibm/2 1-312 (18) 

Also, since Epm = IpmZi., if Epm has doubled but 
Ipm has not changed it can only mean that Zl has 
doubled. In other words, if the oscillator is de- 
signed to operate class A when Re is a maximum, 
it will operate class B when a crystal unit is in- 
serted that has an effective resistance equal to 
Rem/2. Equation (16) can be generalized to apply 
for all operating states in which 2 £ma is equal to 
or greater than E<.o. For greater simplicity, 0 
should be replaced by («■ — $) / 2 , where = *■ — 20 
is the angle during which the tube is cut off. $ is 
always positive, whereas 0 would be negative in 
the case of class-C operation. Thus, equation (16) 
can be expressed 


(all classes) Pzt = 


Eco Ibm [2ir — e + sin d) 


4x^1 -t- cos 


1—312 (19) 


156 



The slope of this equation is positive for all values 
of B less than 2*, so that the power dissipated in 
a crystal unit always becomes greater as R, be- 

comes smaller. By substituting cos for sin ^ in 
equation (11) and rearranging, we have 


Ep„ = E„/(i + co 8|) 


so that 


2 P 

Ipn, = = 2 Pz. I 1 + COS 

"nm 


(■ 


Ibm (2ir — -f sin 0) 


2t(i + cos|) 


1 )/^ 

1—312 (20) 


Equation (20), unlike equation (19), has a maxi- 
mum when 6 is approximately 3jr/5. That a maxi- 
mum (or a minimum) occurs between 8 = 0 and 
0 = IT is to be expected, since Ip™ has the same 
value for each of those values of B. This maxi- 
mum is 


(max) Ip™ = 0.54 Ib™ 1-^12 (21) 

Now, 

Z,, = 2 Pz../ (Ip™)* 1—312 (22) 


On substituting equations (10) and (20) in (22) 


Section I 
Crystal Oscillators 

which could have been predicted on the basis of 
equation (2). Assume that equation (25) holds, 
what will be the value of 6 when a crystal unit 
having a practical minimum value of R. equal to 
Re™/9 is connected in the circuit? When R, = R„, 
the negative term within the parentheses of equa- 
tion (24) is equal to —1; with R, = Rem/9, the 
same term is reduced to —1/9. Thus, the maximum 
0 to be expected is defined by : 

((max) 0 for (min) R,] when: 9 — sin 0 = 16ir/9 

1—312 (26) 

Figure 1-133 shows that equation (26) requires that 

(max) 0 = 16t/9 - 1 1—312 (27) 

In other words, when a sharp-cutoff tube is used 
and the oscillator is designed for class-A operation 
with crystal units of maximum R,, the oscillator 
will be operating class C, with the tube cut off 
approximately three-fourths of the time, when 
crystal units of minimum values of R, are con- 
nected in the circuit. Equation (24) can be gen- 
eralized to define 6 with reference to any conven- 
ient value of R,, simply by assuming that 6 = 0 
when R, = (ref) R,. Thus, 

9 - sin i9 = 2ir [1 - R„/(ref)Re] 1—312 (28) 


or 


Zl = 


2tE. 


Ibm (2 t — 9-1- sin 9) 


1—312 (23) 


Rearranging and substituting l/4<»’C,’Rp for Z,,, 
where C, is the specified load capacitance of the 
crystal unit. 


9 — sin 9 = 2ir (1 — 4u*C, Eco R./Ib™) 


1—312 (24) 


Equation (24) is quite significant in that it pre- 
dicts the approximate angle during which a given 
tube will be cut off for a given value of R,. A 
Pierce oscillator designed so that the tube is oper- 
ating with a class-A bias equal to Eco/2 when R, 
is a maximum will have a value of 9 equal to zero. 
Thus, when R, = R™, each side of equation (24) 
must vanish. For the right-hand side to equal zero, 

4a)* C, Eoo Rem = Ibm 


This is equivalent to saying that 


1 ^ 4 

(min) Zl (min) PI 


= (average) g™ 

ii^co 

1—312 (25) 


9n - sin 9 n = 2ir (N - 1)/N 1—312 (29) 

where N = (ref) R./Rej,, 9 = 0 when R, = (ref) 
Re, and 9 n is the value of 9 for the particular value 
of Re symbolized by Re,,. The reference R* need 
not be the maximum permissible R,. For a given 
oscillator of Cg/Cp ratio equal to 1, (ref) R, would 
be the value of Re that would cause the peak-to- 
peak excitation voltage to equal Eeo in magnitude. 
Assuming that (ref) R, = Rem, what then will be 
the ratios of Pz,, and I,™, corresponding to mini- 
mum and maximum values of R*? When 9 = 

— , as given by equation (27), cos 9/2 is 

very nearly —2/3, so equation (19) becomes 

(Class C max) Pz^ 

_ ~ ~9~) _ Eeo Ibm 

4x(l-|) ^ 

1—312 (30) 

On comparison with equations (7) and (17), 
which give values of Pz,. of EeoIbm/8 and EeoIbm/4 


WADC TR 56-156 


157 



PIERCE OSCILLATOR OPERATION AS A FUNCTION OF CRYSTAL Rf. 0 • ANGLE DURING WHICH TU8E IS CUTOUT. 

/under conditions of no load , sharp-cutoff TUBE,\ IF; 8 ■ 0, WHEN crystal R, IS MAX. PERMITTED, 

UaRGE Ro.AND Co/Cp*t I then: e«-e, when R, - (MAX) R,/N; 


Saction I 

Crystal Oscillators 



NIS 


WADC TR 56-156 


158 


Figure 1-133. Sine curve plotted to tame scale as the angle B. 45-degree diagonal that intercepts sin Bi- on curve intercepts (dk—sin dj on 8 axis 




SacKon I 
Crystal Oscillators 


for clasa-A and class-B operation, respectively, we 
find that where there is to be no output control, 
the no-load tube voltage must be so chosen that a 
crystal unit of maximum R, is not driven at more 
than one-fourth the rated drive level, otherwise 
crystals of small R, will be overdriven. With a 
power ratio of 4 when the Zl ratio is 9, ,it can be 
shown quite simply that the Ip,„ ratio is 2/3 and 
the Epn ratio is 6. I'hus, 

(Class-C min) Ip„ = Ibm/3 1—312 (31) 

and 

(Class-C max) = 3 Eco 1 — 312 (32) 


V2 P„„/4 _ Vp;7~ 

2a)C.V 2wC a/ 2 R„ 


1—312 (34) 


where Pcm is the true drive-level rating. Since 
(max) Ec will also be equal to Ece/2, approxi- 
mately, then (no longer continuing to treat as 
a magnitude only) 

(max) = - VT^ / V"2T57„ 

1—312 (35) 

At the same time, Pzl ^ ^ must not ex- 

ceed Po„/4. Consequently, 


The plate dissipation in the tube should be of 
little concern unless subminiature tubes are used. 
In any event the plate dissipation is a maximum 
when Re is a maximum, so no thought need be 
taken for other than class-A operation. Approxi- 
mately, 

(Class-A) plate power = Ebib = EbIbm/2 

1—312 (33) 

Finally, the foregoing equations suggest that a 
Pierce oscillator employing a sharp-cutoff tube be 
designed for class-A operation on the assumption 
that Re will be a maximum and that the maximum 
permissible drive level is one-fourth its actual rat- 
ing. Under these assumptions, equation 1-293(2) 
should be changed to 

(max) Ee = 


(max) Ib„ = - , = 2<cC. 

(max) I Eee I 

1—312 (36) 

Equations (35) and (36) define the operating 
characteristics to be sought if a sharp-cutoff tube 
is to be used under conditions of maximum output 
for maximum stability. Remember, that equation 
(34) actually is an expression of the limitation on 
I„ the crystal current, and therefore upon Ep and 
E,. As far as the self-excitation voltage of a truly 
sharp-cutoff tube is concerned, it will be difficult 
to keep this voltage from building up until it 
reaches into the bend near the cutoff point. For 
this reason, the first consideration is that Ip™ is 
not exceeded. As a safety measure, Ii,m should not 
be greater than the value given by equation (36), 
even if the actual E™ is less than (max) E,„. The 
conclusions reached in the foregoing discussion 
are summarized in the following table. 


PIERCE-CIRCUIT OPERATING LIMITATIONS DUE TO CRYSTAL SPECIFICATIONS OF 
LOAD CAPACITANCE, C,, MAXIMUM PERMISSIBLE EFFECTIVE 
RESISTANCE, R,^.., AND DRIVE LEVEL, P.„ 

Plate Dissipation (max) = Ezlbm/Z 

E,o = — Pcm/ “Ox y/ i R,ni 

Ibm 2 P<,m/| Eco | 

Conditions are those for sharp-cutoff tube, negligible load, 
gridleak bias with large R^, C^C, = 1, Ep >> Ep, R,, 
> > (max) PI/4 = (max) Zl, Class-A operation when 
R^ is maximum, and maximum permissible output. 

R. = 

Rem 

R«n/2 

R«o/9 

Ep„, Ep„, 1 Ec 1 = 

1 Ec./2 1 

|Ec.| 

3|Ec.| 

Pc (= Pzp) 

P.a./4 

Pco./2 

P™ 

Ipm 

Ibn./2 

Iba./2 

Ilini/3 

Zb 

(min) PI/4 

(min) PI/2 

9 (min) PI/4 

gm — 

4/(min) PI 

2/ (min) PI 

4/9 (min) PI 

6 = 

0 

IT 

ISw - 

9 ~ 

Operation = 

Class A 

Class 6 

Class C 


WADC TR 56-156 


159 








Sadion I 

Crystal Oscillator* 

Pierce-Oscillator Design Considerations When 
Tube Cutoff Has Below-Average Sharpness 

1-313. Unless a vacuum tube has plate character- 
istics resemblinsr those of subminiature tubes 
when normal plate voltagres are used, or unless by 
reducing the filament voltage such characteristics 
can be achieved, a Pierce oscillator tube must be 
operated at a plate voltage of from one-half to 
one-fourth normal. In so doing, it is very probable 
that the lower bend of the EJo curve will become 
rather extended compared with the straight por- 
tion to the left of zero grid volts. In this event, the 
tube exhibits the characteristics of a remote-cut- 
off tube, except that the cutoff voltage is one-fifth 
or less that of a normal remote-cutoff tube operat- 
ing at an equivalent reduced plate voltage. Where 
the cutoff is not sharp, it is quite easy for equilib- 
rium to be reached with peak-to-peak excitation 
voltages much smaller in magnitude than Ecm and 
considerably greater ranges in R, of the crystal 
unit can exist before cutoff is reached. Thus, in the 
more usual case, the assumptions used in para- 
graph 1-312 cannot be made unless greater care is 
taken in the oscillator design to ensure a peak-to- 
peak excitation voltage equal to |E„| when R, is a 
maximum — an operating point much more difficult 
to locate and critical to maintain when a large 
steady decrease in the effective gm occurs well be- 
fore the cutoff point is reached, and which may 
require very low plate voltages if the maximum-R, 
crystal unit is not to be overdriven. As a concrete 
example, suppose that the oscillator is to employ 
a 10-mc crystal unit of the CR-18/U type. At this 
frequency, P<.m = 5 mw, R«„ = 25 ohms, and 
Cl = 32 /»/if. On substitution in equation 1 — 312 
(35), we obtain a (max) of approximately 
— 5V. By equation 1 — 312(36), this value of Ec® is 
to be obtained in a tube where the zero-bias, with- 
out-signal plate current is Ibm = 2 ma. Such char- 
acteristics are not easily obtained with conven- 
tional-sized vacuum tubes. It may be necessary to 
operate at the given value of Ibn» or slightly 
greater, and a cutoff voltage that is of a smaller 
magnitude than that indicated for (max) E„ in 
equation 1 — 312(35) , in which case all crystal units 
used will drive the tube beyond cutoff. An alterna- 
tive approach is to operate at a larger than maxi- 
mum E„, but, if this be done, a safety factor should 
be allowed by assuming that Ip is to be the same 
for all values of R®. Although this will not be 
strictly true, the assumption is a close approxima- 


tion if the change in plate current between the 
values of E® = Eco/2 and E®® is very small com- 
pared with the change in plate current between 
E® = 0 and E® = E,.®/2. It can be seen that insofar 
as the effective Ip can be assumed to remain con- 
stant, Ep, and hence E*, I,, E®, and the crystal 
driving power, VR» =« Ip’Zl, increase directly with 
Zl, or inversely with R®. The problem is to find the 
maximum permissible E®, which, although apply- 
ing to extended-cutoff operation only when R® is 
a maximum, will not lead to a replacement crystal 
being over-driven if its resistance is less than the 
maximum. Again we assume a minimum R® equal 
to R®n,/9. In a manner similar to the derivation of 
equation 1 — 293(2), we can say (max) E® (with 
(min) R®) = 


(max) E® (for (min) R®] = 

2 Re,n 


V2P®.® 

2wCx\/ (min) R® 

1—313 (1) 


Now, if equation (1) gives the bias voltage when 
a crystal unit of minimum R, is connected, assum- 
ing that Ip is constant, the bias that exists when 
a crystal unit of maximum R, (= 9 (min) R®) is 
substituted will be one-ninth the value above. Thus, 


(extended cutoff max) E® [for R® = R,m] = 

1—313 (2) 

3a;Cx %/ 2 Rem 


If a gridleak Pierce oscillator is not to have a 
loaded plate circuit, nor an adjustable nor con- 
trolled output voltage, nor a sharp cutoff, equa- 
tion (2) gives the maximum bias that can be safely 
assumed when R® is a maximum. The output volt- 
age agreeing with equation (2) is two-thirds that 
given in paragraph 1-312 for a sharp-cutoff tube. 
If Rg is not large enough for the average E® to ap- 
proximate the peak excitation voltage, a maximum 
bias less than that given by equation (2) must be 
assumed. With large values of R,, |E®| of equation 
(2) is the peak of the maximum excitation voltage 
when R® is a maximum, and |E®| of equation (1) 
is the approximate peak when R® is a minimum. 
If a Cg/Cp ratio other than 1 is used, equation (2) 
can be expressed more exactly 


(extended cutoff max) E® = 


WADC TR 56-156 


160 



— ! S«C, VTJ;; 1^13 (3) 

It should be understood that although equations 
(2) and (3) are derived from equation (1), it is 
wiser to select the vacuum tube and plate voltage 
upon the assumption that the resistance of the 
crystal unit is a maximum rather than a minimum. 
Since the effective amplification factor of the tube 
cannot be expected to be constant for all values 
of R„ equations (1) and (2) will not both hold for 
the same circuit. If (1) is correct, (2) will indicate 
a value too low; if (2) is correct, (1) will indicate 
a value too high. Equation (2) therefore permits 
a safety factor in the event of an exceptionally low 
value of Also, if the oscillator performs prop- 
erly with R, a maximum, it will almost certainly 
operate when R. is a minimum. The reverse is not 
necessarily true. 

1-314. Equation 1 — 313(1) is equivalent to a bias 
and output of the same magnitude as that obtained 
in paragraph 1-312 for sharp-cutoff conditions and 
minimum R« ; but the bias and output of equation 
1 — 312(2) for R, = Rm, when E„ is assumed to 
be significantly greater than 2Ec, are only two- 
thirds their equivalent sharp-cutoff values. In the 
case of the 10-mc CR-18/U crystal unit dis- 
cussed in paragraph 1-313, the (practical max) Ec, 

as given by equation 1 — 313 (2) is — « —1.7 V. 
For the smallest values of R., the bias will ap- 
proach —15 V. Assuming that gm ^ (according 
to equation 1 — 312(2), when C,/Cp = 1) and that 


Zl 


(min) PI 
4 

10 ‘® 


4 X 6.28* X 32* 


_ 1 ^ 
" 4a)* C/ R.„ 

— = 2600 ohms 

X 25 


then, gni 10V2500 = 400 ^unhos when R, is a 
maximum. This is a very small transconductance 
to be obtained with a bias of approximately —1.7 
volts, and usually cannot be obtained at all with 
normal operating voltages except in the case of 
the small battery-operated tubes. The 1.7-volt 
nuiximum bias represents a peak-to-peak excitation 
maximum of 3.4 volts. With an average g„ of 400 
)>mhos, the limiting value of Ibm (<«> 2Ipn 2g,aE,m) 
becomes 1.4 ma, approximately. Only if age is used 
to provide a much larger bias than can be obtained 
with a peak-to-peak excitation of 3.4 volts will it 
be possible to have such a small zero-signal plate 
current without operating conventional tubes at 
greatly reduced voltages. Generally, it is easier to 
operate with a small E,, and a larger Im and not 


WAOC TR 56-156 


161 


SecHon I 
Crystal Osdliotort 

attempt class-A operation. A large percentage of 
the crystal oscillators now in use drive the crystal 
units at a considerably higher level than is ad- 
vantageous from the point of view of stability and 
long crystal life. Much of the care otherwise taken 
in the circuit design can be wasted if the first 
consideration is power output rather than fre- 
quency control. Where a vacuum-tube manual 
recommends a particular voltage of power ampli- 
fier for use as a class-C oscillator tube, the typical 
operating characteristics listed are rarely appro- 
priate for military-standard crystal units, but 
apply more usually to LC circuits. The plate volt- 
ages must be considerably lower than the typical 
values indicated, in order to reach the small trans- 
conductances that must exist at equilibrium with- 
out overdriving the crystal unit. 

1-315. Assume that a crystal unit is connected in 
a Pierce circuit using a conventional triode oper- 
ating at its normal plate voltage, and that the 
C,/Cp ratio is near unity. The equilibrium values of 
gm and Rp cannot be reached until the amplitude is 
great enough for the tube to be operating class C, 
and the crystal unit will almost certainly be over- 
driven. There are four ways in which the circuit 
can be adjusted to prevent this overdrive: (a) the 
plate voltage can be reduced, (b) the filament volt- 
age can be reduced, (c) the C,/CV ratio can be in- 
creased, or (d) the load losses can be increased. Of 
these methods, the first, reducing the plate voltage, 
seems to be the best from the point of view of 
frequency stability, although a reduction of the fila- 
ment voltage may be worth consideration. Very 
possibly, if the filament voltage is decreased suffi- 
ciently to lower the zero-bias transconductance to 
as much as one-fifth its normal value, the oper- 
ation of the circuit will become unduly sensitive 
to slight fluctuations in the filament power supply. 
The only data available at this writing is that re- 
ported by Messrs. Roberts, Novak, and Goldsmith 
of the Armour Research Foundation of Illinois 
Institute of Technology. Experimenting with a 
6C4 tube and a 7-rnc Miller circuit, it was found 
that a 30-percent decrease in filament voltage, 
which is equivalent to decreasing the filament 
power by approximately one-half or more, depend- 
ing upon the temperature coefficient of the filament 
resistonce, caused only a 2.5-cycle rise in frequency. 
(In a Pierce circuit the frequency would have de- 
creased.) This effect on the frequency is very 
slight, but the exact decrease in the r-f plate cur- 
rent is not known. Nevertheless, the evidence is 
sufficient to suggest that if the tube character- 
istics are made suitable for a crystal circuit by 
reducing the filament voltage, any instability 



S«ction I 

Crystal Oscillators 

caused by further fluctuations in the filament volt- 
age would appear primarily as variations in the 
output voltage, rather than as variations in the fre- 
quency. In view of the fact that a reduction in 
filament current permits a greater saving in power 
than does a reduction in plate voltage (and length- 
ens the tube life), this approach to the problem 
may well be worth experimentation. The conven- 
tional approach, however, is to operate with a low 
plate voltage. If a C*/Cp ratio on the order of unity 
is to be used, the average tube will require plate or 
screen voltages of 40 to 50 volts, or less. The lower 
the voltage, the nearer class-A operation can be 
approached at equilibrium. A fair approximation 
of the operating conditions to be expected can be 
made from an inspection of a family of plate-char- 
acteristic curves. With Cg/Cp = 1, the peak-to-peak 
variations in plate voltage are the same as those 
of the excitation voltage, so for all practical pur- 
poses the plate voltage can be assumed to be con- 
stant. Thus, the load line can be assumed to be 
vertical, and the maximum and minimum ampli- 
tudes of Ip for a given plate voltage become the 
values, respectively, for grid voltages of 0 and 
2Egm, where Eg^ is the peak excitation voltage. For 
the 10-mc crystal unit taken as an example above, 
it was found that the peak-to-peak I, for a maxi- 
mum Re was 2 Igre Eel = 1.4 ma. The correct plate 
voltage for a given tube is thus the value of Ep at 
which a change of grid voltage from 0 to —3.4 
volts causes the plate current to decrease by 1.4 
ma. This type of operation — class A to class AB — 
is generally more feasible when age is used, if it is 
desired to apply for all values of R,. 


PIERCE-OSCILLATOR DESIGN CONSIDERA- 
TIONS FOR Cg/Cp RATIOS OTHER THAN ONE 


1-316. When the Gg/Cp ratio is not approximately 
equal to one but the total load capacitance meets 
the crystal specifications, gm is increased, and gen- 
erally it will be easier to obtain desirable vacuum- 
tube characteristics at more convenient plate volt- 
ages. The first step, as before, is to theoretically 

limit the peak of the crystal current to * / ^ 

, . \ Rgm 

When Re is a maximum. The peak excitation volt- 


1 rsp 

age, Ega, equals “TT^/ “ 5 ^ under these conditions. 

Ep,„ equals equals C./CpZi,: I„m 

equals gmEgp,. ^ith these values taken as a start 
we can retrace the steps taken in paragraphs 1-312 
and 1-313, and determine the values of Ipm and Ec® 
that do not permit the crystal to be overdriven 
for any value of R, between R«a and R,o,/9. 


FINAL WORD ON CORRECT LOAD 
CAPACITANCE IN THE PIERCE CIRCUIT 

1-317. A prime purpose of the military specifica- 
tions regarding the load capacitance, effective re- 
sistance, drive level, and frequency tolerance of the 
different types of crystal units is to guarantee the 
replacement of a defective crystal unit in the field 
without special testing or other complications, and 
with the same ease that a defective vacuum tube 
can be replaced with a new tube of the same type. 
However, a crystal unit is more critical in its per- 
formance than a vacuum tube. As a result there 
can be no replacement guarantee unless the new 
crystal unit is inserted in a circuit where it will 
be operated under approximately the same load 
and drive conditions at which it has been tested. 
An inspection of the various types of oscillator 
circuits now in use, such as those illustrated in 
figures 1-135 to 1-138, most of which have been 
designed around the older types of crystal units, 
reveals a much greater versatility in operating 
conditions than is now desired in the design of 
new equipment. One of the requirements that is no 
longer within the jurisdiction of the design engi- 
neer is the effective load cajiacitance into which 
the crystal unit is to work. This means, that for 
a given nominal frequency and type of crystal unit, 
the crystal unit must exhibit a given inductive 
reactance, X*, equal numerically to l/<oC„ where 
C, is the rated load capacitance. Furthermore, it 
means that for each particular crystal unit there 
is but one frequency at which it is supposed to 
operate. This does not mean that all crystal units 
of the same type and nominal frequency have a 
single common operating frequency, rather that 
each has its own individual frequency, which, how- 
ever, will not differ from the nominal frequency 
by more than the permitted tolerance. It is the 
effective operating reactance that the crystal units 
must have in common. Approximately, 

X, = — 4irLAf — (Equation (1), figure 1 — 98) 

, , 4irLm 

Xco 

Now, Af = f, — f„ where fp is the operating par- 
allel-resonant frequency and f, is the series-reso- 
nant frequency of the motional arm. Assume that 
a 10-mc parallel-resonant crystal unit has a fre- 
quency tolerance of ±0.02 per cent. This is equiv- 
alent to an absolute frequency tolerance of ±2000 
cps. Two crystal units at opposite extremes could 
be within specifications even though their oper- 
ating frequencies, fpi and fp„ were 4000 cps apart. 


WADC TR 56-156 


162 



If the rated load capacitance were 32 and the 
crystals were A elements, Af, itself, for each crys- 
tal would be on the order of 2000 cps. If the 
crystals were 6 elements of the same shunt ca- 
pacitance, Co, Af for each crystal would be only in 
the neighborhood of 800 cps, because of the B ele- 
ment’s larger series-arm inductance, L. It becomes 
obvious that there can be no expectation of “pull- 
ing” the frequencies together by making slight 
adjustments in the load capacitance. The lower- 
frequency crystal could not be raised to zero beat 
with a frequency 4000 cps higher without reduc- 
ing C. several-fold. The higher-frequency crystal 
could not even be “pulled” to the nominal frequency 
and oscillations still be maintained. For this reason, 
the design engineer should generally not attempt to 
provide an operator with frequency adjustments 
for the crystal oscillator. The only adjustments 
needed are those which can be factory preset, 
in order to compensate for slight differences 
in stray capacitance. If the frequencies to be gen- 
erated must be in close agreement with some 
standard, or with the frequency of some control- 
ling station, the task is to provide oven-controlled 
crystal units of smaller tolerance. Only when the 
desired operating tolerance is less than any pro- 
vided by crystal-unit specifications alone, is it 
necessary to provide the operator with a frequency 
adjustment knob. Even then, the adjustment need 
not provide a tuning range greater than the 
specified crystal tolerance. Since the smaller toler- 
ances are only 1/10 to 1/20 of the 0.02 per cent in 
the example above, the total variation in load ca- 
pacitance may not need to be greater than ±10 
per cent of the specified capacitance. 

1-318. It may be desirable to provide an operator 
with the means of controlling the output voltage 



Figure J-134. Ganged eapaeitances to enable adjuai- 
menf at C,/C, ratio at Pierce circuit wHhaut changing 
fatal load capacitance 


SocNon I 
Crystal Oscillators 

of a Pierce oscillator by varying the Cg/Cp ratio. In 
this case, care must be taken to ensure that the 
total load capacitance remains the same. If Cg and 
Cp are to be adjusted separately, some type of 
matching scales should be provided with the two 
tuning knobs, so that the correct load capacitance 
can always be had when, say, the two scales give 
the same reading. It is more desirable to have 
available ganged capacitors similar to those shown 
in figure 1-134 for each of the Military Standard 
capacitance ratings. The capacitors C, and Cj in 
series are to be designed to always ensure a correct 
load capacitance when each is shunted by conven- 
ient predetermined fixed capacitance. The small 
variable capacitances C, and C« are adjusted until 
the sum of their values and the circuit stray capac- 
itances, C.I and Cn, provide the correct fixed 
shunts for the ganged elements. 

MODIFICATIONS OF PIERCE CIRCUIT 

1-319. Figures 1-135, 1-136, 1-137, and 1-138 and 
their accompanying circuit-data charts reveal a 
great flexibility in the design of a Pierce oscillator. 
It would be very convenient to be able to put our 
finger on a single circuit and say that the design 
of this circuit is superior to all others. Unfor- 
tunately, this is not possible. One would first have 
to define what is meant by “superior design.” The 
definition, at best would be a complex function of 
several physical and psychological variables. The 
very existence of a wide variety of circuit modi- 
fications suggests that no one circuit is superior 
to all others for all given tasks, although much of 
the variety can be attributed to the desire of the 
design engineer to create his own circuit and also 
to avoid the risk of possibly infringing upon the 
patent, rights of another. Our space does not per- 
mit a detailed discussion of each of the circuits 
shown. Only a few of the highlights are to be 
mentioned. In general, most of the oscillators illus- 
trated employ older-type crystal units ; most of the 
circuits use B-f voltages on the order of 200 volts, 
and would overdrive the smaller-sized crystals cur- 
rently recommended; and the load capacitances 
and the Cg/Cp ratios vary widely from one circuit 
to another. Those circuits that employ currently 
recommended Military Standard crystal units 
(crystal units having nomenclature type numbers, 
CR-15/U and higher) ' are designed to operate so 
that the crystal unit faces its rated parallel-mode 
load capacitance. In these circuits the plate sup- 
ply is normally 100 to 120 volts, and the crystal 
unit of average resistance is operated at 4 to 5 
milliwatts. 


WADC TR 56-156 


163 




Circuit Data (or Figure 1-135. F in kc. R in kilohma. C in iiitf. L in nh. 

WAOC TR 56-156 164 


















































165 






































Saclien I 

Crystal Oscillalort 



FIgun 1-135. Ceirfinwod 



Fig. 

Equipment 

Purpose 

F, 

F, 

F, 

CR 

R, 

R« 

R. 

(L) 

Receiver-Trans- 
mitter RT-173/ 
ARC-33 

“Side-step” 
injector oscil- 
lator 

7662 . 5 

104,840- 

192,280 

F.-l-Pj 

CR-18/U 

47 

68 

68 

(M) 

Receiver-Trans- 
mitter RT-173/ 
ARC-33 

Main channel 
local oscillator 

12,517.8 

15,325 

im 

BfH 

11 


33 


Lear Radio Set 

Model T-30AB- 

RCBBI^2 

Local oscillator 

655- 

7155 

455 

IF. 



'68 

0.68 

5 




4755- 

3845 


l■EOl 

FT-243 

0.33 

47 

150 

(P) 

Radio Set AN/ 

VRC-3 



4300 

(1st I.F.) 

■REsI 

mi 


0 33 

47 

150 

(Q) 


Local oscilla- 
tor 

1175- 

8175 

1700- 

8700 

Fe-F, 

(525 

I.F.) 


60 

0.4 

250 


Circuit Data for Figure 1-135. F in kc. R in kilohms. C in pjtf. L in jih- 


WADC TR 56-156 


166 















































WADC TR 56-156 


167 







































Circuit D*ta for Figure 1-lSS. F in kc. R in kiiohms. C in L in ^h. 

WADC TR 56-156 16S 





































S*cHon I 
Cryttal 0»cillat«r« 



R. 

R. 


C. 

C. 

C, 

wm 

■1 

a 

L, 

mm 

V. 

v. 



24 

100 

3,000 

m 




500 


1/2 

12AT7 











2000 






B 

27 

470 





iimii 


■■ 




27 


n 





2000 


■UiMI 




20 

830 

3000 

6 




500 


■HRH 




m 

m 


1-8 




500 





■■ 

155 

15 








MOPM 

■UPS 

BQS2BI 



22 

Hi 

1500 

820 




RFC 


m 


■ 

■ 

■ 

220 

1500 

1500 

2 

1500 

4t 



'A 5670 

■ 


WADC TR 56-156 


169 





































Section I 

Crystal Oscillators 


TO FRCO MULTIPLIER 


SC4 


>4 I 


P4 TO FREO D(VI0ER[(A)] 

TO FREO MULTIPLIER [(8)] 


FI 

462.4SKC 



(C) B-f 

figure 1-136. ModiHcaiions of Pierce oscillator using screen-grid tubes 


Equipment 

Purpose 

F, 

Radio Set AN/ 
FRC-10 (WECo 
Transmitter 
D-156000) 

L-F osc for 
single-side- 
band opera- 
tion 

625 

Same as (A) 

H-F osc for 
single-side- 
band opera- 
tion 

940- 

5000 

Switching 

Unit SA-107 
( )/MRC-4 

BFO for two 
diversity re- 
ceivers 

462.45 

Radio Receiver 
R-270/FRR 

BFO 

462.45 

Radio Receiver 
BC-659-( ) 

Local osc of 
receiver 

5675- 

8650 


R4 Rs 


20 0 


500 1000 


1000 270 


Circuit Data for Figure 1-136. F in kc. R in kilohms. C in mmI. L in fih. 


WADC TR 56-156 





















































CT 



( 0 ) 


FI 

(6‘9MC) 


CSCR 


ftgvr* 1-136. ConHnutd 



c , 

Cj 

Ca 

c , 

c . 

C, 

Cr 

100 

10,000 

10,000 

10,000 

1000 

■ 

50 

Vari- 

able 

10,000 

10,000 

10,000 

1000 

■ 

0 

100 

10- 

100 

100,000 

75 

20,000 



39 

10,000 

100 

100 

95 

100 

5 

1000 

10,000 

55 

10,000 

25 




WADC TR 56-156 171 



























































S«eli«ii I 

Crystal Oscillators 



Fig. 

Equipment 

Purpose 

n 

B 

CR 

R, 

R; 

R. 

o 

R. 

R. 

Rr 

(F) 

Radio Trans- 
mitter BC- 
329-N 

M.O. 

200- 

400 

■ 

FT-249 

100 

0.22 

15 

10 

66 

5 

0.022 

(G) 

Lear Radio 

Set T-30AB- 
RCBBL-2 

M.O. 

2900- 

6500 



470 

0.1 

100 

100 

■ 

18 

■ 

(H) 

Modulator- 

Transmitter 

T-233/URW-3 

M.O. (or 
remote-con- 
trol trans- 
mitter 

3966 6- 
5666 6 

2Fi 

CR-IA/ 

AR 

47 

0.33 

6 8 

30 

27 

0 0051 

■ 

(I) 

Modulator- 

Transmitter 

BC-1158 

M.O. 

3966.6- 

5666 6 

2Fi 

CR-IA/ 

AR 

47 

0.33 

6 8 

30 

27 

0 0051 

■ 

(J) 

Radio Trans- 
mitter 12GLX-2 

M.O. 

260- 

1750 

F, 

Holder 

FT-249 

100 

0.1 

50 

385 

B 

■ 

■ 


Circuit Data for Figure 1-136. F in kc. R in kilohma. C in mmI- L in uh. 


WAOC TR 56-156 172 





































































WADC TR M-156 


Section I 
Crycfal OtdHotore 


r3-*n OR «ri 



c . 

C , 

n 


L . 


V, 

■ 1 , 

10,000 

10,000 

10,000 


18,000 

18,000 

6V6 

GT/G 

807 




■ 



6U 

6L6 

45 

470 

470 




1/2 

815 

1/2 

815 

45 

470 

470 




1/2 

815 

1/2 

815 

100,000 

100,000 



Per 

F, 


807 

807 


173 












































Section I 

Crystal Oscillators 



Figure I -137. E/ectron-coup/ed Fierce otclllator modMeations. All circuit* 
except circuit (A) provide frequency multiplication 


Fig. 

Equipment 

Purpose 

F, 

Fs 

F, 

CR 

Ri 

mm 

Rs 

R< 

R. 


Receiver-Transmit- 
ter RT-i78/ARC-27 

Spectrum osc- 
illator 

10,000 

D 


CR-27/U 

100 

56 

12 

100 

0 

(B) 

Radio Receiver 

R-252/ARN-14 

2nd monitor 
osc of 

receiver vfo 

11,275- 

11,725 

M 


■jm 

8.2 

68 

■ 

■ 


(C) 

Radio Trans- 
mitter BC-400- 

CB,C,D,E) 

M.O. for mar- 
ker beacon 

4166.67 

2Fi 


FT-164 

200 

35 

5 

20 

0 

(D) 

Radio Trans- 
mitter T-67/ 

ARC-33 

M.O. 

5555- 

8666 

2F, 

6Fi 

CR-IA/AR 

5.1 

H 

47 

47 

0.083 

(E) 

Radio Set 
AN/ARC-IA 

Main-cbannel 

heterodyne 
freq osc 


2Fi 

18F, 

CR-IA/AR 

CR-18/AR 

100 

8.2 

■ 

100 


(F) 

Frequency Meter 
TS-186/UP 

Crystal 

calibrator 

5000 

1 

1 

Navy CG- 
40210; GE 
*G31 Ther- 
mocell. 

6L6 tube 
envelope; 
heater; ± 
0.002 per 
cent, —20 
to 75*C 

1 

100 

100 

10 

100 

(G) 

Frequency Meter 

TS-186(B/C)/UP 



Crystal 

calibrator 

5000 

2Fi 

■ 

CB-18/U 

(oven, 

60*0 

1.5 


100 

10 

H 


Circuit Data for Figure 1-137. F in kc. R in kilohms. C in ttitl. L in <4h. 


WADC TR 56-156 174 









































































WADC TR 56-156 


175 
























































































Section I 

Crystal Oscillators 





Fig. 

Equipment 

Purpoee 

F, 

F, 

F, 

CR 

R. 

R. 

R. 

m 

■a 

(H) 

Aircraft Radio 

Corp. Radio 
Transmitter ARC 
Types T-13 and 
T-llA 

M.O. 

6000- 

9000 

3Fi 

1 

CAATC 

11081 

100 

0.56 

56 

0.18 

100 

(I) 

Radio Receiver 
R-270/FRR 

Local oscUla* 
tor 

1565= 

8511.6 

Fi, 

2F„ 

or 

3Fi 

■ 


47 

27 

■ 

■ 

■ 

H 

Radio Trans- 
mitter BC-655- 

a,-am 

M.O. 

5560- 

8660 

1 

1 

DC-ii- 
( ). DC- 

16, DC- 
26. or 

CR-1( )/ 

AR 

50 

50 

0.05 

50 

1 

(K) 

Radio Transmitter 
BC-401-B 

M.O. 


jj^ll 



100 

0 51 





Circuit Data for Figure 1-137. F in kc. R in kilohma. C in nitt- L in «•)>. 


WADC TR 56-156 176 








































vt 



Hgun I- 137. CanMnuMl 


R. 

Rt 

R. 

C, 


MM 

B 

1 


1 

1 

S5 

120 

120 

40 

(T-13) 

50 

(T-11) 



■ 

■ 

100 

9-140 

10,000 

10 

1 


■ 

■ 

10 

50 


6800 

6 




250 

6000 

6000 

6000 

6 


WADC TR 56-156 


Svdion I 
Crystal Otcillatorf 


F2 



F2 VZ 



■ 

B 

a 


B 

B 

B 

mm 

50 

100 

750 

■ 

16,900 


6AQ5 

6AQ5 

■ 

■ 

■ 




m 


1- 

5.5 


■ 

■ 

2500 

(60fl) 

9-1/2 

turns 

H 

H 

000 

6000 

1000 

350 

2500 


m 

HI 


177 







































Section I 

Crystal Oscillators 



Circuit Data (or Figure 1-137. F in ke. R in kilohms. C in mit. L in jih. 

WADC TR 56-156 '7B 

























































FIgwi* 1-137. CouHnomd 


R. 

1 

Ri 

wm 

a 

m 

■ 


■ 


25 

10,000 

3000 

6-42 

(eaci 

sect- 

ion) 

100 

■ 


25 

100 

3000 

6-42 

(eacl 

sect- 

ion) 



■ 

20 

220 

2000 

10 

0 1 

27 

28 

33 

565~ 

100 

10,00 


■ 

■ 

1000 

10 


15 


WADC TR 56-156 


Section I 
Cryttol Otdilators 


R7 




(P) 


1 

D 

c. 

C: 

ID 

D 

D 

D 

V. 

li 

1000 



■ 

1700 

(40fi) 

■ 

6SK7 

9003 

1 

1000 



■ 


9.3 

■ 

9003 


8000 

90 

103 

5 

500 


H 


0 

3 

10,000 

1.5-7 




5639 

Balanced 

modulator 


220 

10,000 

60,000 


500 

■ 

mmn 



179 






















































Section J 

Crystol Oscillators 


R3 



(0) (R) (S) 


FI 

I3.02S KC 



CR‘I6/U 


FIgvn f-f37. ConflniMd 



Fig. 

Equipment 

Purpoae 

F. 

D 

(Q) 

Radio Set 

AN/ARC-34(XA-1) 

2nd monitor osc 
of transmitter 
M.O. 

3230-3900 

(5 

crystals) 

F, 

(R) 

Radio Set 

AN/ARC-34(XA-1) 

3rd monitor osc 
of transmitter 
M.O. 

3650-3775 

(4 

crystals) 

Pi 

(S) 

Radio Set 

AN/ARC-84(XA-1) 

4th monitor osc 
of transmitter 
M.O. 

5130-5165 

(6 

crystals) 

Pi 

(T) 



13,025 

m 

(U) 

Radio Receiver 
R-322/ARN-18 

Local osc 

11,490- 

11,700 

(20 

crystals) 


(V) 

wm 

Coarse fre<^ 
osc for mixing 
with output of 
fine freq oac 

4800-9600 

(28 

crystals) 

3Fi 


Circuit Data for Figure 1>137. F in kc. R in kilohms. C in fifd, L in 
WAOC TR 56-156 180 


ISO V 


•H30 V 


V2 



SUPPRESSOR 
GRID IN (Q) 
ANO (R) 
CONTROL 
GRID IN (S) 


TO 



(T) 


D 

CR 

R. 

Ri 

R. 

R. 

R. 

■ 

CR-27/U 

560 

56 

B 

1.8 

0.68 

■ 


560 

56 

m 

1.8 

() 

■ 

gnu 

560 

56 

m 

1.8 

0 



560 

0.56 

■H 



■ 


0.01 

■ 

B 

■ 

■ 


CR-18/tJ' 

100 

0.15 

100 

■ 

■ 


Mh. 






















































- ' ' ! Section I 

•» v"’-,-,'.; Cryitol Ascillcrtert 



•I 


n 

S4.#T0-3S,I00KC 



R. 

Rt 

Ri 

a 

C, 

D 

B 

D 

MM 

D 

D 

L, 

a 

V, 

V, 

1 

0.47 

10 

47 

1000 

10 

33 

10,000 

470 

■ 

22 

5000 

■ 

5840 

5636 

0 

■ 

10 

47 

1000 

10 

33 

10,000 


12 

0 

5000 


5840 

6636 

0 



■! 


H 

■n 

10,000 

470 

10 

M 



5840 

6840 


m 



22 

■■ 

10,000 

10,000 

1000 

130 


500 


5840 



■ 

■ 

6 


■ 




■ 

■ 


m 




■ 

■ 

■ 

6 

■ 

■ 

m 

■ 

1500 

■ 

500 


6AK6W 

1 


WAOC TR 56-15« 


181 


















































Section I 

Crystal Oscillators 


F2 C3 n 



IB) 


Figure 1-138. Flectron-coupled Pierce oscillator modifications for heterodyne circuits 


Fig. 

Equipment 

Purpose 

F, 

Fa 

Fa 

CR 

Ri 

R2 

(A) 

Radio Receiver R-262/ARN-14 

1st monitor osc 
of receiver vfo. 

14,000- 

20,500 

From iscH 
lation 

amplifier of 
vfo. 

n 

CR-33/U 

10.4 

10 

(B) 

Signal Generator TS-413/U 

Crystal harmonic 
heterodyne mixer 

1000 

Variable 

rf 

ISIS 

igim 

DC-9-AJ 

1000 

35 

(C) 

Radio Receiver R-146A/AKW-35 

2nd heterodyne 
osc 

5456 

5000 


FT-243 

47 

47 

(D) 

R-F Signal Generator Set 
AN/URM-25C 

Crystal calibrator 
harmonic generator 
and mixer 

1000 

10-50,000 
(output from 
variable osc) 


CR-18/U 

1 

270 

(E) 

— 

Radio Receiver R-277 (XA-A)/ 
Al’N-70 

Local osc 

400 

480 

2850 

2950 

3000 

3050 

F.-Fs 

300 

HOO 

(IF) 

CR-26/U 

(LF,) 

CR-18/U 

(HF.) 

22 

0.068 


Circuit Data for Figure 1-138. F in kc. R in kilohms. C in fi/if. L in uh- 

WAOC TR 56-156 182 
















































Section I 
Crystal Oscillators 


FS 



Rs 

R4 

Rs 

D 

C^ 

Ca 

C, 

Cs 

c. 

L, 

U 

La 

V, 

47 

22 

6.8 

68 

27 

470 


47 

1800 

■ 

1 


5725/ 

6AS6 

100 

10 


330 

5600 

10,000 





1 


6SA7 

47 

47 


25 

5-30 

55 

50,000 






12SA7 

30 

100 

510 

1000 

6-25 

270 

10,000 

10,000 


2500 

500 


6BE6 

22 



47 

470 

100 

47,000 






5750 


WADC TR 56-156 


183 



















































section I 

Crystal Oscillators 

Preventing B+ Voltage from Existing 
Across Crystal Unit in Pierce Circvit 
1-320. There are two commonly used methods for 
preventing the application of the B+ voltage 
across the crystal unit. One method is to connect a 
blocking capacitor in series with the crystal unit. 
The other method is to connect the crystal unit 
directly to ground, and the plate to ground through 
an r-f bypass capacitor. With this arrangement, 
the cathode must be operated above r-f ground. 
Except for the advantages of electron-coupling to 
the load to be had in the tri-tet circuit, which uses 
screen-grid tubes, the r-f-grounded-plate modifies^ 
tion is not to be preferred since the plate-to-grid 
capacitance directly shunting the crystal unit is 
greatly increased by the addition of the grid-to- 
ground capacitance. When a capacitor is used to 
block the B+ voltage from the crystal, a number 
of modifications are possible, three of which are 
shown in figures 1-135 (F) , (G) , and (K) . In each 
figure, the blocking capacitor is the one labeled Cj, 
and has a reactance negligible with respect to that 
of the crystal unit. The more usual circuit arrange- 
ment is that shown in figure 1-135 (K). In figure 
1-135 (G), Cj plays a dual function in blocking 
the B+ voltage from both the crystal and the 
grid circuit of the succeeding stage. The circuit 
in figure 1-135 (F) , although not effective in block- 
ing the B+ voltage from the crystal unit, does, of 
course, reduce the d-c potential across the crystal 
equally as well as the other arrangements. It is 
desirable to keep the d-c potential across the crys- 
tal unit low ; otherwise the crystal may be heavily 
strained in one direction, and the chance is in- 
creased that the elastic limit of the crystal will be 
approached on the alternation of the a-c voltage 
of the same polarity, or, at least, that the effect 
on the crystal will cause the performance char- 
acteristics to deviate from test specifications. Since 
the crystal unit, itself, is a capacitor of consider- 
ably greater dielectric thickness and smaller cross- 
sectional area than the usual blocking capacitor, 
it might seem questionable that the d-c voltage 
across the crystal unit could be expected to be 
significantly reduced. Certainly in the static state, 
an air-gap crystal unit should have at least as high 
a resistance to leakage currents as would the block- 
ing capacitor. However, in the dynamic state some 
degree of ionization will occur if the voltage is ex- 
cessive. The consequent leakage tends to charge 
the blocking capacitor to the full plate-to-grid d-c 
voltage. An unchecked B+ voltage not only can in- 
crease the likelihood of corona effects, but also can 
lead to continuous discharge should corona losses 
once begin, and to an increase in the effective re- 


sistance of the crystal unit, a reduction of the grid 
bias, and perhaps even to arcing and puncturing 
of the crystal. A blocking capacitor, if it does not 
remove the d-c potential completely from across 
the crystal unit, at least can ensure that the po- 
tential is not sufficient to cause ionization. A block- 
ing capacitor is generally more important in 
high-drive circuits employing air-gap crystal units. 

1-321. Figure 1-135 (X) shows one example of a 
grounded-crystal, grounded-plate Pierce circuit. 
At first glance such an oscillator might very easily 
be mistaken for the Miller type. C* is a relatively 
large capacitance that causes the plate to be at r-f 
ground. Ci is effectively Cp, the plate-to-cathode 
capacitance, and Ci serves as the lumped part of 
Cp. By interchanging Ci and C, and making the 
ground connection of the crystal a plate connec- 
tion, essentially the same oscillator characteristics 
are obtained except that the B+ is across the crys- 
tal unit. Note that the circuit is designed for the 
maximum possible output voltage, in that the out- 
put is taken across the crystal unit directly, in- 
stead of across Ci, the effective plate capacitance, 
alone. The fact that the plate is at r-f ground does 
not remove the grid-to-plate capacitance from 
shunting the crystal unit, but adds to this the grid- 
to-ground capacitance. In the circuit of figure 
1-135 (R), a small tuning capacitance, C,, is also 
shunted across the crystal unit. In addition to this 
there is the extra shunt capacitance contributed by 
the oven in which the crystal is mounted. In cir- 
cuit (R), the Ci/Cj ratio, which is approximately 
equal to the C*/Cp ratio, is on the order of %. The 
gridleak losses are increased somewhat, since the 
grid resistance is connected across the crystal in- 
stead of across the grid capacitance. All these 
factors tend to reduce the effective Qf of the feed- 
back circuit, so it would appear that with crystal 
units of greater than average resistance the cir- 
cuit operates with the tank considerably detuned 
from resonance. The fact is, however, that con- 
necting the grid resistance across the crystal 
serves to concentrate most of the grid losses in the 
plate-to-grid circuit and to eliminate them from 
the grid-to-cathode circuits, and this probably in- 
creases the effective feed-back Qf more than the 
increased losses diminish it. The circuit in figure 
1-135 (X) is similar to that in figure 1-135 (R) 
except that the tuning capacitance has been elim- 
inated and the output is obtained directly across 
the crystal unit. It is claimed that this arrange- 
ment tends to smooth the output and to reduce 
the harmonics. What probably is meant is that 
for a given output voltage the harmonic content is 
less. This can readily be seen, for the voltage across 


WADC TR 56-156 


184 



the crystal unit is equal to the sum of the voltages 
across Ct and Ci. If the same output is to be taken 
across either capacitance alone, the excitation 
must be increased and the circuit will generally be 
operated beyond tube cutoff a greater fraction of 
the time, thereby increasing the hig^r-harmonic 
content. It should also be noted that the output 
arm in figure 1-135 (X), since it shunts the crys- 
tal, is effectively part of the feed-back circuit. 
Should the load increase or decrease, so also will 
the excitation. The circuit is thus a tri-tet modifi- 
cation where the output voltage is somewhat 
stabilized against changes in the load, but only at 
the sacrifice of frequency stability. This feature 
is not important in the particular fixed-load cir- 
cuit of figure 1-135 (X), since the load in this 
circuit appears to be reasonably constant. 

Electron^oupled Pierce Oscillator 
1-322. The electron-coupled oscillator permits a re- 
markable freedom from coupling between the plate 
load circuit and the crystal circuit. Screen-grid 
tubes are required, with the screen grid serving as 
the plate of the oscillator circuit. When electron 
coupling is used in conjunction with Pierce oscil- 
lators, the tri-tet arrangement, where the plate 
load circuit is in series with the oscillator tank, 
is generally the most advantageous, and is used 
in all the circuits shown in figure 1-137 except in 
circuits (I) and (0). With the screen at r-f 
ground, the vacuum-tube plate circuit performs as 
a conventional pentode r-f amplifier, with the ex- 
citation of the control grid a function of both 
the screen and plate r-f currents. Variations in the 
plate impedances have much less effect upon the 
frequency than do similar variations in the oscil- 
lator tank impedances. For this reason, even if the 
oscillator is to operate over a wide range of crystal 
frequencies, a tunable coil and capacitor tank can 
be placed in the plate circuit to obtain a smoother 
sine-wave output without running the risk of 
greatly changing the load capacitance of the crys- 
tal circuit. In effect, the electron-coupled oscillator 
reduces by one the number of amplifier stages that 
are required, and hence is particularly applicable 
for small portable transmitters where the crystal 
circuit must perform as nearly as possible the func- 
tion of a power oscillator. The widest application 
of the electron-coupled Pierce circuit is for the pur- 
pose of frequency multiplication. In figure 1-137 
(D), for example, the plate tank circuit is tuned 
to twice the crystal frequency. The L/C ratio of 
the plate tank should be as small as practicable, in 
order to increase the output selectivity and to en- 
sure a low-impedance bypass through the coil for 
the fundamental frequency and through the ca- 


Section I 
Crystal Osdllalors 

pacitor for all harmonics higher than the second. 
The larger the angle 6 during which the tube is 
cut off, the larger will be the percentage of the 
higher-harmonic generation in the output. In gen- 
eral, the lower the order of the harmonic, the 
greater is its energy content. For optimum output, 
the tube should not be heavily conducting during 
a positive alternation of the plate harmonic volt- 
age, Eh. During such intervals the plate tank would 
be losing energy to the circuit at an instantaneous 
rate of ibOb, where ib is the instantaneous d-c plate 
current and eb is the instantaneous harmonic volt- 
age across the plate tank. To meet the require- 
ments above, plate current should be allowed to 
flow only during the interval of approximately one 
alternation of a harmonic cycle. Since the tube is to 
be cut on and off at the fundamental frequency, the 
plate tank, after receiving a pulse of energy dur- 
ing one alternation of a harmonic cycle, must 
oscillate freely for the remaining part of the funda- 
mental period. If the frequency is being doubled, 
plate current should flow approximately one-fourth 
the time; if the frequency is being tripled, plate 
current should flow approximately one-sixth of the 
time. To generalize, if the frequency is to be multi- 
plied n times, optimum n’th harmonic output is 
approached if the oscillator is designed so that the 
tube conducts approximately l/2n of each funda- 
mental cycle. In paragraph 1-312, it was found 
that for a given peak value of ib, (Ibm) , the effective 
Ip was essentially constant for all bias voltages be- 
tween class-A and class-B operations, although a 
small maximum occured when the tube was cut 
off during three-fifths of the negative alternation. 
See equation 1-312 (20). If the plate load is con- 
stant, as is the case in the electron-coupled circuit, 
this point of maximum Ip coincides with the condi- 
tions of maximum output. If we assume that ap- 
proximately the same conditions hold in the case 
of frequency multiplication, maximum harmonic 
output is approached if the tube continuously con- 
ducts 7/10 of the period of one harmonic cycle, or 
7/lOn of the period of the fundamental cycle. Since 
the maximum is not at all sharp, the optimum 
operating conditions are not critical and can be 
assumed to extend over a range which permits the 
tube to conduct from l/2n to 7/lOn of the time. 
That is, for optimum output, the oscillator section 
can be designed so that the crystal unit of average 
resistance allows the tube to be cut off during a 
fundamental-cycle angle within the range given by 

- 1 ) to 

n 5n 

1—322 (1) 


WAOC Tt 56-156 


1«5 


S«ction I 

Cr/ctal OscillQtQrs 

The principal advantage of the tri-tet circuit is 
that the excitation voltage tends to increase with 
an increase in load. If the output is to be inductively 
coupled to the succeeding stage, the tri-tet ar- 
rangement tends to stabilize the output voltage 
when the coefficient of coupling is varied. This fea- 
ture originally found its greatest popularity among 
radio amateurs, although it was used principally 
in conjunction with Miller rather than Pierce oscil- 
lators. Figure 1-139 shows the basic tri-tet circuit 
as applied to Pierce and to Miller oscillators. It is 
the adaptability to frequency multiplication rather 
than to variable load conditions, however, that is 
of greatest importance when considering the tri- 
tet circuit for use in militarj equipments. The tri- 
tet frequency stability is low compared with that 
of the conventional pentode circuit because of the 
large stray capacitance (approximately 8 ^t^f ) that 
directly shunts the crystal unit. About 4 mti is the 
of the tube, and the remainder is the capac- 
itance of the grid leads to ground, which otherwise 
would be part of C,. 


Miscellaneous Pierce Circuit Modifications 
1-S23. Circuits (A) and (B) in figure 1-135 are 
of interest because they indicate two stages in the 
development of a particular modification of the 
Pierce oscillator. Originally Ls, C„ C4, and R4 were 
not present. Since no grid capacitance is employed 
other than that of the tube, the C,/Cp ratio is very 
small. The tube has approximately 4 md capac- 
itance between plate and grid, and it may be as- 
sumed that the crystal oven adds a comparable 
amount directly across the crystal. In all proba- 
bility the phase-shifting Q, of the feed-back circuit 
is rather low, so that the tank can be expected to 
be more reactive than resistive. L, is resonant with 
Cl at the mid-point of the intended frequency 
range. If the feed-back circuit is operated with a 
high effective Q*, this value of L, should provide 
a zero phase shift in Ip. In this event, the tube would 
operate into a resistive load, and a theoretical in- 
dependence of the frequency with changes in Rp 
could be predicted. As it is, the low feed-back Q, 


lA) 

PIERCE TRI-TET 


( B) 

miller tri-tet 

Figure 1-139. Basie tri-tet circuits where excHaflan voltage Is a function of both screen- and 

plate-circultr-f currents 





WADC TR 56-156 


186 



probably requires the crystal tank circuit to be de- 
tuned to such a point that at equilibrim it appears 
either as a reactance much greater in magnitude 
than that of the coil L,, or as a reactance approxi- 
mately equal to that of Ci, in which case the lagging 
component of the current through the crystal unit 
is negligible compared with the current through 
Cl. Under these latter conditions, the tube would 
operate into a low-impedance, series-resonant cir- 
cuit at the mid-point of the frequency range, where 
Xli + Xoi = 0 .- Nevertheless, it was found that 
oscillations could not be maintained dependably at 
the mid-point of the frequency range. It was for 
this reason that the changes were made in the 
models represented by circuit (B) , as indicated in 
the data chart for figure 1-135. The tank circuit 
L 3 C, is resonant at the mid-point of the frequency 
range. C, has been changed so that the reactance 
of the coil L, is lower than the reactance of the 
crystal tank at all frequencies. R 4 has been added 
to dampen the effect of L, at the high end of the 
frequency range. It may be that the dead spot at 
the mid-point of the frequency range in circuit 
(A) was due only to transient effects in the crystal 
units before oscillations could build up, or it may 
have been due to the fact that the feed-back Qi 
was insufficient to provide the necessary plate im- 
pedance to maintain equilibrium even if oscilla- 
tions were once started. 

1-324. An interesting circuit is that shown in fig- 
ure 1-136 (D). The feed-back voltage is developed 
across C, by the r-f plate current. Cs, although of 
the same capacitance as C 4 , maintains the screen at 
r-f ground, since R, is very large. The large value 
of R, keeps the screen voltage, and hence the out 
put, at very low values, so that the crystal is only 
weakly driven. 

1-325. The circuit shown in figure 1-136 (E) is 


Section I 
Crystal Oscillators 

intended to supply a fourth-harmonic excitation 
of the V» stage. For this purpose the CjLa tank is 
tuned to 4F,. A low Lj/C, ratio is provided, to en- 
sure that the fundamental is effectively bspassed. 
The capacitance C» is kept small so as to present 
a high reactance to the fundamental, else the fun- 
damental would be entirely bypassed around the 
crystal circuit. 

1-326. The circuit shown in figure 1-136 (G) is 
something of a novelty in that a Pierce instead of 
a Miller oscillator is employed to directly excite the 
power amplifier of a small transmitter. The L,C 4 
arm is a neutralizing circuit which prevents the 
amplitude-modulated output stage from varying 
the effective impedance of the oscillator load. Nor- 
mally, neutralizing networks are not necessary for 
crystal oscillators. Only when the oscillators drive 
power amplifiers directly is feed-back neutrali- 
zation advisable. Even then, if the power amplifier 
is not modulated and performs as a frequency 
multiplier, neutralization is not necessary. 

1-327. The electron-coupled converter circuits 
shown in figime 1-138 embody more or less the 
same features previously discussed. The basic 
methods illustrated for obtaining a heterodyne out- 
put are more or less self-explanatory, and will not 
be elaborated upon here. 

The Miller Oseillafer 

1-328. The Miller oscillator is the crystal equivalent 
of a Hartley oscillator in which no mutual induct- 
ance exists between the plate-to-cathode and grid- 
to-cathode inductances. ( See figure 1-140.) The 
Miller oscillator has an average frequency devia- 
tion of approximately 1.5 times that of the Pierce 
circuit. The plate circuit must appear inductive in 
order that the correct phase shift will be produced 
in Ep, the plate r-f voltage, to compensate for the 


t 

Zpg 

\ 


2g 




Figun 
WAOC TR 


(A) 

MILLER OSCILLATOR 


1-140. Diagram* Ulu$tratlng tha agulvalanca oatwaaa th» Millar tirtult and tha Hartley tircuil 
56-15A 187 



Section I 

Crystal Oscillators 


resistance in the feed-back arm, since this resist- 
ance prevents the necessary 180-degree phase 
rotation of the equivalent generator voltage of the 
ampliher from occurring entirely in the feed-back 
circuit. The effective load capacitance into which 
the crystal unit operates is, approximately. 


C. 


= c. 



1—328 (1) 


where Z„g and Zp are both considered as unsigned 
magnitudes, and the various symbols correspond 
to those in figure 1-140. Since the load capacitance 
is a function of the frequency, a Miller oscillator 
cannot be operated at more than one frequency and 
still present the same load capacitance to each crys- 
tal unit except by providing for an adjustment of 
the circuit parameters. In spite of its greater fre- 
quency instability and lack of circuit simplicity 
as compared with the Pierce circuit, the Miller de- 
sign is the one most widely used in crystal oscil- 
lators. The reason for this popularity is the greater 
output that can be obtained for the same crystal 
drive level. In either the Pierce or the Miller basic 
circuit, the output cannot exceed the voltage across 
Zp„ the largest single impedance in the plate tank 
circuit. In the Pierce circuit, the maximum voltage 
is thus the maximum permissible across the crystal 
unit ; in the Miller circuit the maximum voltage is 
(k -|- 1) times the maximum permissible voltage 
across the crystal unit, where k is the gain of the 
stage, equal to Ep/E*. This gain, theoretically (not 
practically), can approach the mu of the tube as 
a limit when the load impedance, Zl, is large com- 
pared with Rp. Thus, the use of a Miller circuit 
permits a saving of one amplifier stage. 

1-329. The feed-back capacitance of the Miller cir- 
cuit is, normally, simply the plate-to-grid inter- 
electrode capacitance of the tube. It cannot, of 
course, be less than this unless an inductive shunt 
is connected between the plate and grid. When a 
pentode is used, it is usually necessary to insert 
a small feed-back capacitance on the order of a few 
micromicrofarads. The waveform in the output is 
improved by the use of a tuned tank circuit having 
a low L/C ratio in place of Lp. The plate tank must 
be tuned to a frequency above the oscillator fre- 
quency, in order that the tank impedance will ap- 
pear inductive. Such an arrangement also ensures 
a large effective Lp of high Q. A variable capac- 
itance in the plate tank facilitates adjustments to 
obtain the correct load capacitance for the crystal 
unit. 


MILLER-OSCILLATOR DESIGN 
CONSIDERATIONS 

1-330. If it is decided to employ a Miller oscillator 
as a frequency generator, the choice should be 
dictated by the need of a greater output than can 
be obtained with a Pierce oscillator. An exception 
to this rule might be made if a tri-tet circuit is 
contemplated, in which case, the large capacitance 
that will directly shunt the Pierce-connected crys- 
tal may well prevent the stability from being as 
high as that of the Miller tri-tet circuit. The Miller 
circuit is the more critical to design insofar as 
maintaining the correct load capacitance is con- 
cerned, but the basic approach to the problem is 
the same as that which was followed in analyzing 
the equilibrium state of the Pierce circuit. Both 
oscillators are represented by the same basic cir- 
cuit, shown in figure 1-119. We shall not repeat the 
steps involved in the derivation of the equilibrium 
equations in the particular case of the Miller oscil- 
lator. The basic equations given in the following 
paragraphs can be used as points of departure in 
the design of any Miller circuit. Also, by methods 
similar to those employed in the analysis of the 
Pierce circuit, the design limitations of a Miller 
oscillator in which the crystal unit is to be oper- 
ated within specifications can be predetermined, 
approximately. 


MILLER-OSCILLATOR EQUATION OF STATE 

1-331. As in the case of the Pierce circuit, there 
are two equations that express the state of oscil- 
lation equilibrium in the Miller circuit. Originally 
derived by Koga, these two equations are the real 
and the imaginary parts of the general equation: 


MZpZg 

Rp Z, -p Zp(Zg Zpg) 

where 


1-331 (1) 


Z, = Zg -|- 2^ -4- Zpg 
and 

Zg = Rpg -f- jXgj Zp = jXp; Zpg = jXpg. 


R,,, is the effective resistance of the grid circuit, 
accounting for both the crystal and gridleak losses. 
The losses in the plate circuit are assumed to be 
negligible. On solving equation (1), the real part 
can be expressed as 


^ Xp [Z,^ (M + 1) + X, Xpg] 

Reg (Xp -t- Xpg) 


WADC TR 56-156 


188 


1—331 (2) 



Equation (2) defines the conditions that exist 
when the feedback power input equals the power 
dissipated in the grid circuit. The imaginary part 
of the equation (1) defines the frequency, or, more 
exactly, the impedance relations that must exist 
if the feedback is to be of proper phase. This is 
given as 


X, = 1-331 (3) 

Where is equal to X»/R,.g, and X, = X,, + X* -f 
X,«. Equation (1) is the same 'Ss equation 1-289 
(1) except that the terms are rearranged and ^ 
is substituted for the product R,.g,„. Equations (2) 
and (3) correspond to equations 1-289 (2) and (3) , 
respectively. If it is assumed that Z, >=« X„ and 
that (Xp, -|- Xp,,,) != —X,,, equation (2) above can 
be simplified, thus; 


R>. 


Xp, [X, (m + 1 ) + Xp„l 
R..„ 


1—331 (4) 


Remember that Xp. and X^ are positive, and that 
X,.,, is negative. When equation (3) is rearranged 
as follows 

Xp. + X, -F = - Xp. 

1—331 (5) 


it can be seen that the effect of the tube Rp on the 


frequency is a function of the term 



LOAD CAPACITANCE OF CRYSTAL UNIT 
IN MILLER OSCILLATOR 


1-332. The load capacitance into which the crystal 
unit operates in a Miller circuit has been derived 
by Koga to be 


C. = Cp. -t- Cp. -I- Cv 1—332 (1) 

where C. and C,,. are as represented in figure 1-140, 
and 


Section I 
Crystal Oscillators 

latter equation in the following form : 

T + ^ + —7— R r = 0 

■ " + 

1—332 (3) 

It is next assumed that the term in parentheses 

1 -I- Rp/Zp -f — = 1 -f Rp/Zp 

1—332 (4) 

Such an assumption not only implies that the leed- 
back current is negligible compared with the r-f 
current through the plate coil, L,,, but that the 
feed-back impedance is so high relative to Rp that 
Ri,/(Zk -f- Zp,,) is negligible compared with 1. The 
former implication requires that Z,., the effective 
load impedance across the tube, be approximately 
equal to jX,,; the latter implication requires that 
X,,p,| — IXp,! >> Rp. If the effective phase-deter- 
mining Qf of the feed-back circuit is 10 or more, as 
is very likely to be the case when standard crystal 
units are operating at their rated local load ca- 
pacitance and Cp. is not excessive, then Ep must be 
very nearly in phase with — mE.. Such a condition 
cannot exist simultaneously with equation (4) un- 
less Rp, < < Xp — an operating state that would be 
very undesirable from the point of view of fre- 
quency stability. If R,, is to have a reasonable 
value at the rated load capacitance of the crystal 
unit, the impedance of the feed-back arm cannot be 
greatly different from that of the plate circuit. 
Equation (4) would be sufficiently accurate for 
very low values of Q. and very large transconduct- 
ances for the tube; however, it would seem that 
for crystal units that are to be operated well above 
series resonance, the approximation of equation 
(4) should not be made. In this case 

1 -t- Rp/Zp + „ = 1 + Rp/Zl 

"g “T ^pg 

'1—332 (5 


It appears that equation (2) gives a value to Cr 
that, for a Miller oscillator operated at the rated 
load capacitance of the crystal unit, is probably 
between three and four times too small for the 
average circuit. Equation (2) is derived from 
equation 1-331 (1), beginning by expressing the 


and Zi, approaches Xp.^/R,.. as R,, and Q. increase. 
Since Zt., as used above, represents an involved 
complex quantity, an exact expression of equation 
(2) will not be attempted here. As can be seen 
from equation 1-331 (5), if 1/Q. and X„/Rp are 
each on order of 1/10 or smaller, X. -f Xp, =» 1X,,.|. 
On the other hand, if Xp,/R,, is not small, the vari- 
able parameters of the vacuum tube and the varia- 
tions to be expected in the effective resistance from 


WADC TR 56-156 


189 



Section I 

Crystal Oscillators 


one crystal unit to another will have such a large 
influence upon the effective load capacitance that 
there can never be an assurance that a crystal 
chosen at random will be operated according to 
specifications. In other words, a Miller oscillator 
cannot be designed to provide approximately a 
specified load capacitance unless X* + X,, |X,,,|. 

Under these conditions 

C, = C„ -1 — = C* + Cp, + Cx„ 

1—332 (6) 

where 

It must be understood that equations (6) and (7) 
assume that R,, is large compared with X,„ and that 
Xg is large compared with R. ^.. For this latter con- 
dition to hold, the grid-to-cathode capacitance, Cg, 
must be kept as small as possible. If the assump- 
tions above cannot be made, it is not feasible to 
expect a Miller oscillator to operate at approxi- 
mately the same load capacitance for all crystal 
units, nor can good frequency stability be ex- 
pected. A more comprehensive capacitance equa- 
tion for the Miller circuit — one that holds approxi- 
mately for all operating conditions — can be ex- 
pressed as 

C, = Cg -I- Cpg -h Cxp' 1-332 (8) 

where Cx,.' is given by equation (7), except that 
X|, is replaced by X,,', where 

Xp' = Xp + R.g/Qg - 1^2 (9) 

It will be seen that X,,' has been so chosen that 
equation 1-331 (5) can be expressed in the form 


where P„ is the power dissipated in the output 
circuit, and 


= EgVP* 


1—333 (2) 


where P^ is the power dissipated in the grid cir- 
cuit. If the gridleak losses are negligible, P,, equals 
the crystal power and Rg,. equals the PI of the 
crystal unit. As can be seen from the equations in 
figure 1-141, either PI must be small or R, very 
large for this assumption to hold. With Rp large 
compared with Zj,, Ip is approximately equal to 
gmEp. We shall assume that X, = X^ -f Xp + X,* 
« 0. Under these conditions it can be shown quite 
simply that in the circuit of figure 1-141, letting 

k = Ep/Eg, 


k" R„ + Rp 
k R„ R„ 


1—333 (3) 


For a given Rg,. and Ro, equation (3) has a mini- 
mum g,„ when 


k* = Ro/Rgg 1—333 (4) 

Since a maximum Rp coincides with a minimum 
gm, equation (4) also establishes the conditions for 
a maximum Rp. Now, Rg, is a function of R, of the 
crystal unit, so that a circuit design using equa- 
tion (4) should be based on a most probable value 
of Rg, (i.e., a most probable value of R,), which 
will usually correspond to a value of R, between 
one-third and one-fourth of the maximum R,. 
Equation (4) should not be interpreted to mean 
that if Ro/Rg, is adjusted to equal a fixed value of 
k*, the g„, of the tube will therefore be a minimum 
relative to its values for other R,/Rg, ratios. Such 
an interpretation would only hold true if the prod- 
uct R„Rg, were constant. Where equation (4) 
holds, it can be shown that the ratio of output 
power to crystal power is 

P„/Pg = 1 1—333 (5) 


Xg -h Xp' + Xpg = 0 1-332 (10) 

In the event that the plate circuit contains a ca- 
pacitance shunting the coil, equations (6) and (7) 
still hold except that Xp refers to the total parallel 
reactance in the LpCp branches. 

MAXIMUM Rp OF MILLER OSCILLATOR 
TUBE UNDER GIVEN LOAD CONDITIONS 

1-333. Referring to figure 1-141, Rp and R*, are 
defined as follows ; 

Rp = EpVPp 

WADC TR 56-156 


OPTIMUM VALUE OF k - EJE, 

FOR MILLER OSCILLATOR 

1-334. Practical values of k, unless a pentode is 
used, are limited by the plate-to-grid and grid-to- 
cathode interelectrode capacitances of the tube and 
the specified load capacitance of the crystal unit. 
If X. = Xg -f- Xp -f Xpg « 0, then 

Xp = kXg 1—334 (1) 

and 

1—334 (2) 


1—333 (1) 


190 


- Xp, = (k + 1) Xg 



Section I 
Crystal Oscillators 



figure U141. Equivalent circuit of Miller oscillator. Up is assumed to be large compared with the total load 
impedance. Il„ is an equivalent resistance accounting for the output losses. R,„. is an equivalent resistance 
accounting for the crystal and grid losses; it is approximately equal to the resistance of the parallel circuit 
shown in (Bi. PI is the performance index of the crystal unit; K„' is the equivalent grid resistance; and R„ 

is the actual gridleak resistance 


By equation (2) 


. — (Xp* + X,) 

x;^ 


where 



1—334 (3) 


X. = 



— l/«Cpg , 


Cx = rated load capacitance, and = grid-to- 
cathode capacitance. With triodes, values of k 
above 4 or 5 are difficult to obtain. If the oscillator 
is to be^ designed with no other feedback-circuit 
capacitances than those provided by the interelec- 
trode capacitances, 0,.* and C^, of the tube, it can 
be assumed that k is a fixed parameter equal to 
the value given by equation (3). The output arm 
must thus be designed to provide a reactance, 
Xp =« kXg, if the crystal unit is to operate into its 
rated load capacitance. 

1-335. From the point of view of frequency sta- 
bility it is desired that the term (XpXpe/RpQ*) in 
equation 1 — 331 (5) be as small as possible rela- 
tive to (X, 4- Xp), or, equivalently, to IXpgj. In 
other words. 


WADC TR 56-156 


should be a maximum. With equation 1 — 331 (4), 
it can be shown that 

Rp/k = Q,X,(m - 1) 1-335 (1) 

Equation (1) indicates that as the fraction of the 
loop reactance (kR,,,/Rp) dependent upon Rp be- 
comes smaller, the effective amplification factor of 
the tube becomes greater. Intuitively from equa- 
tion (1) it can be seen that with Q^X^ constant for 
a given vacuum tube and plate voltage, Ei„ R,, must 
increase as k is made larger, otherwise ^ could 
not decrease. Nevertheless, the larger that k be- 
comes the smaller the value of Rp/k. If k = 1, a 
frequency stability almost approaching that of the 
Pierce circuit can be achieved, but with twice the 
output voltage. Lower values of k would soon de- 
tune the oscillating tank to a point where the 
simplifying assumptions made regarding k would 
no longer hold. 

1-336. Since the principal purpose of using a Miller 
instead of a Pierce circuit is to eliminate an am- 
plifier stage, and since Ep is limited to the maxi- 
mum voltage that can be placed across the crystal 
unit, the ratio Ep/E^ = k can be chosen to give 
a desired gain over that which would be o''tained 
with a Pierce oscillator operating at the same crys- 
tal drive level. Ej in the Miller circuit caa-be as- 
sumed to be twice the Eg of a Pierce circuit that 


191 



Saction I 

Cryttol Oscillator* 

has a k = 1. If an imaginary gain of 10 is desired, 
k for the Miller circuit should be equal to 5. For 
k to be 5, according to equation 1 — 334 (3) 

Cp* = (C, - C«)/6 

If Cx —32 fifit and Cf = 8 C,* must be 4 ji/if, 

which is a value quite representative of the aver- 
age triode amplifier, or which could be obtained 
with a pentode by using a small external plate-to- 
grid capacitance. Equation 1 — 335 (1) can be re- 
written in the form 

l/Rp = g,n - Rc«/kX,= 1-336 (1) 

For k = 5, the value of l/R,,, and hence the per- 
centage effect of R|, on the loop reactance, will be 
a minimum the more nearly that g,„ can be made 
to approach in value R.^/kX^* = R,.„/5Xg'. If the 

X 

effective Q of the crystal unit, Q, =; is equal to 

10 or more and if the gridleak losses are negligible, 
it can be shown that 

and that 

X, = 1-336 (3) 

Ag -f- A(;^^ 

Thus, 

Q, = 1_336 (4) 

Re Xc* 

and 

X,VRc« == XeVRe = PI 1—336 (5) 

where PI is the performance index of the crystal 
unit. If it is further assumed that the output losses 
are negligible, the impedance of the crystal tank is 

Y 2 lf2 Y 2 

Z,. = ^ = -- ■! ■ = k^ PI 1—336 (6) 

Reg Reg 

Equation ( 1 ) can thus be written 

1 _ _ 1 _ _ k 

Rp k PI Zl 

1—336 (7) 

If k is fixed by output considerations, the percent- 
age effect of R„ upon the loop reactance becomes 
a function of R,. alone, being a minimum when 
R„ is a maximum. If R,, is increased without limit, 
g,„ approaches k/Zi, as a limit, and the greater 
the PI, the larger will R,. become. In the case 


of the Pierce oscillator, it will be recalled that 
a maximum R,, was obtained by a proper choice 
of k. This optimum k was the one that provided 
the maximum excitation voltage. In the Miller 
circuit, the excitation is the voltage developed 
across the crystal unit, and thus is limited by 
the crystal specifications regardless of the value 
of k. The smaller that k is made, the smaller will 
be the effective R,„ but, even so, the percentage 
effect of R„ upon the effective loop reactance will 
also be smaller. With k fixed by the requirement 
to eliminate an amplifier stage, the problem of ob- 
taining a maximum R,, becomes one of keeping the 
load requirements to a minimum, selecting the 
vacuum tube, determining the proper operating 
voltages consistent with the crystal specifications, 
designing a test model accordingly, and experi- 
menting for optimum results over the resistance 
range to be expected in the crystal units. 

OPERATING CONDITIONS OF MILLER 
OSCILLATOR PROVIDING MAXIMUM R, 
FOR GIVEN g„. 

1-337. If the bias of a tube is supplied by age, the 
excitation voltage is small by comparison, so that 
the operating point of the tube can be theoretically 
estimated by consulting the R,, and g,„ curves 
plotted against grid voltage. The operating bias 
for a given plate voltage w ould approximately be 
that giving values of R,, and g,„ that obey equation 
1 — 336 (7). Unfortunately, there are no curves 
available that indicate the effective R,. and g,„ for 
large excitation voltages where the tube is cut off 
a large fraction of each cycle. Nor has a theoretical 
basis been established for estimating the probable 
rates of change in R,, and g,„ as the excitation is 
increased under various circuit conditions. If time 
permits, experiments designed to furnish such data 
may gain for the engineer a valuable insight into 
the characteristics of his design models. Most 
probably the “dynamic” curves of R,, and g,„ will 
correspond closely to the static curves. Yet the 
possibility exists that significant differences in the 
rates of change in the tube parameters may be 
discovered under certain operating conditions. In 
equation 1 — 336 (7) it can be seen that for any 
large value of R,„ g„, very nearly equals k/Z,.. For 
example, if R,, = 0.5 megohm, the difference be- 
tween g„, and k/Z,, is only 2 micromhos. An R,, of 1 
megohm corresponds to practically the same value 
of g„„ the difference being only on the order of 1 
micromho. Thus, when R,, is large, g,„ can be con- 
sidered more or less a circuit constant. During the 
time that the tube is cut off, R,, is infinite and g,„ 


WADC TR 56-156 


192 



is zero. From the point of view of a large effective 
Rp it is desirable that the cutoff angle be a maxi- 
mum. The larger the g„ of the tube above cutoff, 
the greater can be the cutoff angle. A sharp-cutoff 
tube would be preferred for this purpose. It is also 
desirable to have R,, as high as possible above cut- 
off. For this purpose, a high-mu tube is to be pre- 
ferred. A theoretical estimate of the optimum 
relation between the values of Rp and gm, for a 
tube of the same class-A mu, that provides a maxi- 
mum over-all effective R„ cannot be attempted 
here. However, it would seem that the emphasis 
should be placed upon the larger gm/Rp ratio. The 
effective Rp of a pentode can always be increased 
artificially by inserting a high resistance in the 
plate circuit in series with the oscillating tank, as 
illustrated in figure 1-142. In testing a given tube 
for those bias and excitation conditions which pro- 
vide a maximum Rp, it may be preferable to control 
the bias independently of the oscillations, or by 
using an adjustable, r-f-bypassed cathode resistor. 
A crystal unit should be employed having param- 
eters known to remain constant over the experi- 
mental drive-level range. After the tube has 
warmed up, if the cathode bias is used, the cathode 
resistance can be decreased until oscillations begin. 
The cathode resistance can then be increased until 
the frequency is a maximum. The maximum fre- 
quency would be an indication of an equilibrium 
point of maximum Rp. In order for oscillations to 
be maintained in the event that all the bias is de- 
veloped across the cathode resistance and the ex- 
citation is insufficient to drive the grid positive, a 


Section i 
Crystal Oscillators 

small percentage decrease in the excitation ampli- 
tude must cause at least an equal percentage de- 
crease in the average plate current, and hence in 
the bias. Such operation will require that the tube 
be cut off for a large fraction of each cycle. An 
adjustable cathode resistance cannot be considered 
a particularly practical design feature, but it may 
prove advantageous in an experimental circuit for 
finding the operating conditions that provide a 
maximum Rp for a given g„,. 

FREQUENCY-STABILITY EQUATIONS 
FOR MILLER CIRCUIT 

1-338. Regardless of whether the circuit condi- 
tions are such that the effective load capacitance 
of the crystal unit is assumed to be given by equa- 
tion 1 — 332 (1), by equation 1 — 332 (6), or by 
equation 1 — 332 (8), the fractional change in fre- 
quency for a small change in any one of the equiva- 
lent component capacitances is given by the gen- 
eral equation 


df 1 dC, 

f Fx, ■ C, 


1—338 (1) 


where dC, is equal to dC^, dC,,,,, dC,, dCxp, or dCxp', 
and represents an incremental change in any of the 
component capacitances, and Fx-,. is the frequency- 
stability coefficient of the crystal unit, equal to 
2 Ct’/CCx. (See equation 1 — 243(1).) If a tuning 
capacitor, C,, is connected across Lp in the plate 
circuit, and if equations 1 — 332 (1) and (2) are 
assumed approximately correct, it can be shown 



■Figvre 1-142. The large resistante, R,/, connected in plate circuit effectively increases R,, of the tube. This method 
can be used to improve the frequency stability of a Miller oscillator employing a screen-grid 
tube and an externally connected feed-back capacitance, C,,/ 


WADC TR 56-156 


193 


Sactlon I 

Cryital Oscillators 

that for variations in C„, 

df Cpj 2 n w Cp Rp^ dCp 

“T “ ■ Fxe Xp (1 + RpVXp^) ■ Cp 

1—338 (2) 

where <o = angular frequency. If the circuit is 
operating at maximum activity, in which case 
Zi, = Xp = Rp, equation (2) becomes 

df _ _ Cpg n Rp uCp dCp 

f C. ■ 2Fx. ■ Cp 

1—338 (3) 

Where equations 1 — 332 (6) and (7) can be as- 
sumed to be approximately correct, it can be shown 
that a fractional change in the plate reactance, Xp, 
causes a fractional frequency deviation of 

df _ (Cx Cpg) dXp ^ ggg 

f Cp, Fxe Xp ' 

If no tuning capacitor is provided to shunt the 
plate coil Lp, equation (4) can be expressed in 
terms of a fractional change in Lp, thus: 

^ ^ _ (Cx - Cp,) dLp 1^338 (5) 

f Cp, Fxe L., 

If desired, equations indicating the frequency sta- 
bility when other parameters are varied can be 
derived by following a procedure similar to that 
employed in the analysis of the Pierce circuit. 

MILLER CIRCUIT AS A SMALL 
POWER OSCILLATOR 

1-339. The ratio of the output power to the input 
power is given by the equation 

P„/P, = R,p/Ro 

where k = Ep/E,, and R,,, and R„ are the resist- 
ances represented in figure 1-141. In practice, ra- 
tios of R„. to R„ can be obtained on the order of 4 
for crystal units of maximum effective resistance. 
If k = 5, this would mean a power ratio of 100. 
A 10-mw crystal unit could thus be used to develop 
a 1-watt output. Much higher power outputs, of 
course, can be obtained with crystal units of small 
values of R,, or of higher power ratings. It cannot 
be recomended that a crystal be driven beyond its 
rated power level, but if an exception should ever 
arise, the Miller circuit will require the least over- 
drive. If a larger drive level is necessary than can 
be obtained with Military Standard crystal units, 


the cognizant military agency should first be con- 
sulted. It may be that one or more of the crystal 
manufacturers has available a nonstandard crystal 
unit with crystal dimensions and mounting suffi- 
cient to withstand the required drive — perhaps by 
operating with an overtone mode — without the 
risk of significant parameter variations. As a final 
resort, it will be found that most of the Military 
Standard crystal units can withstand, without 
shattering, drive levels from 10 to more than 20 
times the rated drive. If need be, power outputs 
greater than 35 watts can be obtained with the 
Miller circuit, using a beam power tube or a power 
pentode. It is much easier to obtain a large output 
from a high-mu than from a low-mu tube for the 
same crystal drive. Also, it is easier to obtain a 
large output from, say, a 50-watt tube operated at 
low efficiency, than from a smaller tube operated 
at high efficiency. An r-f choke must be used in 
the grid circuit if large output is to be developed. 
Furthermore, fixed bias that is sufficient to pre- 
vent the grid from drawing current must be used, 
so as to reduce the grid losses to a minimum. The 
voltage gain of the oscillator, k = Ep/E„ should 
be as high as possible. With proper design, except 
that the crystal unit is operating at tolerances 
greater than those specified for low drive levels, 
the Miller circuit can be made to drive a pow'er am- 
plifier of 300 watts or more. Some crystal units can 
withstand as much as 120 ma r-f current and still 
be within the safe-operating range as far as shat- 
tering is concerned. A pressure-mounted unit is 
generally to be preferred at high drive levels, be- 
cause of the added protection it offers, and because 
its greater thermal conductivity permits the gen- 
erated heat to escape more rapidly. For maximum 
output, the oscillator must operate into an im- 
pedance matching the R„ of the tube. If a Miller 
oscillator is to drive a power amplifier, great care 
must be taken in neutralizing the feedback from 
the amplifier, or the crystal may easily be over- 
driven to the point of shattering — that is, unless 
the power amplifier is to serve as a multiplier 
stage, or if a screen-grid tube is used as the amp- 
lifier tube. The plate supply voltage for the oscil- 
lator can be as high as 350 to 650 volts, and that of 
the power amplifier, 1500 to 2000 volts. If a low- 
power (7.5 watts, approximately) oscillator tube 
is used, a fixed bias of 40 to 60 volts will be required 
for high efficiency. A 3.5 to 4.5-ampere current 
in the plate L,.C„ tank can be obtained under these 
conditions. A fixed bias is usually not necessary 
when a 50-watt tube operated at low efficiency is 
used. With the same plate voltage as for the low- 
power tube, a plate tank current of 4.5 to 7.5 am- 


WAOC TR 56-156 


194 



peres can be had. When a fixed bias is used, some 
arrangement must be provided to cut it in after 
oscillations build up and the negative peaks of Ep 
must be sufficient for plate limiting to occur at the 
positive peaks of Ep. When used as a power oscil- 
lator, the Miller circuit is often required to operate 
as a variable-tuned circuit, with a coil or tank- 
circuit in place of the crystal, the circuit thereby 
being converted into a tuned-plate-tuned-grid or a 
Hartley type oscillator. Because of this, the vari- 
ous tuning adjustments and meters that are 
needed in the variable circuit are also available in 
the crystal circuit. In this event, the rated load 
capacitance of the crystal unit will usually exist 
more in theory than in application. Since the crys- 
tal is intended to be operated at high drive, the 
risk is greatly increased that a chance adjustment 
may overload the crystal to the shattering point. 
This risk can be minimized by the use of an r-f 
milliammeter in series with the crystal, with the 
danger zone well marked. Besides excessive plate 
voltage and stray feedback due to poor shielding 
or neutralization, a poorly bypassed screen-grid 
circuit can lead to an overloaded crystal, as also 
can an excessive control-grid bias. Now, no attempt 
should be made to design a circuit in which a crys- 
tal unit is to be operated above its rated power 
level unless weight, space, or expense requirements 
demand the elimination of every possible amplifier 
stage; unless greater frequency stability is re- 
quired than can be obtained with a conventional 
inductor-capacitor network ; or unless a long oper- 
ating lifetime is not a primary consideration. Even 
so, if a Military Standard unit is operated beyond 
specifications, it should be well understood that it 
is no longer effectively a standard type, and no 
guarantee exists concerning the replacement of 
one crystal unit by another. 


Section I 
Crystal Oscillators 

TYPICAL CHARACTERISTICS OF 
MILLER OSCILLATOR 

1-340. Figure 1-143 shows an experimental Miller 
circuit, the performance characteristics of which 
were investigated by Messrs. E. A. Roberts, Paul 
Goldsmith, E. K. Novak, and J. Kurinsky of the 
Armour Research Foundation at the Illinois In- 
stitute of Technology. The crystal units used are 
of the type CR-18/U. The crystal PI indicated for 
each of the characteristic curves (figures 1-144 to 
1-148) of the oscillator in figure 1-143 is the value 
observed when the crystal was operating into its 
rated load capacitance. The PI at the rated load 
capacitance indicates the relative activity of the 
crystal unit, but is not intended to imply that the 
same PI is in effect for all variations of the load 
capacitance. The curves in figure 1-144 indicate 
(excitation voltage)* and the output voltage as the 
plate tuning capacitance is varied. An increase in 
the plate capacitance means an increase in the ef- 
fective value of L„, so that the frequency decreases. 
Thus, as Cp increases (Ca in figure 1-144) , the re- 
actance, X„ of the crystal unit decreases. Oscilla- 
tions cease whenever the Q, of the grid circuit 
becomes too small for the proper phase rotation 
to take place, or the ratio of Xp/X, becomes too 
high for the feed-back voltage to be of sufficient 
amplitude, or the plate tank approaches the paral- 
lel-resonant state, so that E„ can no longer assume 
its proper phase, which requires the plate arm to 
be an inductive reactance smaller in magnitude 
than the capacitive-feedback reactance. The per- 
centage points in figure 1-144 refer to percentages 
of the maximum output voltage that was obtained 
through variations of the plate tank capacitance 
alone. The six curves shown represent values for 



figure 1-143. experimeHtal Miller oeeHIpter whoee charatferletk curvet are plotted In figuret 1-144 ta 1-148 
WADC TR 56-156 195 





Section I 

Crystal Oscillators 

an Eb of 200 volts with a full load of 5000 ohms, 
and for an Eb of 100 volts with full load and with 
no load. It is interesting to note that in each of 
the three pairs of curves the maximum grid volt- 
age occurs at a smaller load capacitance than that 
at which the output voltage is a maximum. For 
each curve where the plate tuning capacitance is 
the same, it can be approximately assumed that 
the load capacitance is the same. Also, between the 
values of C,. = 75 and C„ = 90 iint, it can be as- 
sumed that the percentage change in Ci is small. 
Since the PI of the crystal unit is the same where 
the load capacitance is the same, the excitation- 


voltage-squared curves indicate the relative crys- 
tal drive for the different Eb and load conditions. 
Also note that the two pairs of curves represent- 
ing full-load conditions coincide fairly closely at 
their points of equal percentages. This is impor- 
tant in interpreting the curves in ftgure 1-145, 
each of which represents 50 per cent output, and 
hence approximately the same load capacitance 
and frequency. Exceptions are the lOK curves in 
figure 1-145, as can be checked by figure 1-146. 
The performances curves in figure 1-145 are the 
Miller equivalents of the Pierce curves in figure 
1-130. It can be seen that the Miller output is much 



figure 1-144. (A) Square of exeitation vohaga and (B) rms valum of output voltage vanvs plate tuning capac- 
itance of experimental Miller oscillator, frequency = 7 me; gridleak resistance = I megohm; and PI of CR-IB/U 
crystal unit (with lead capacitaoce of 32 = 49 kllehms. f.L = full load conditions (S,000 ohms across /date 

tank) and N.L. = no load conditions. Percentage points refer to percentages of maximum output voltage 
obtainable under given lead and d-c plate veftago conditions 


WADC TR 56-156 


196 




Saction I 

Crystal Oscillatera 



WADC TR 56-156 


198 



more sensitive to changes in the grid resistance. 
This is to be expected, since the grid-to-cathode r-f 
impedance and excitation voltage is much greater 
in the Miller circuit. A crystal r-f voltage of 2 volts 
represents an excitation of 2 volts in the Miller 
circuit, but usually of only 1 volt or less in the 
Pierce circuit. If the curves in figures 1-130 and 
1-145 were plotted against excitation voltage in- 
stead of crystal driving power they would be much 
more similar in appearance. 

1-341. The frequency curves in figure 1-146 are 
the Miller equivalents of the Pierce curves in figure 
1-129. Note that as R, is decreased the frequency 
falls, whereas in the Pierce circuit the frequency 
increases. This is one reason why the Miller circuit 
becomes so much more frequency sensitive to 
changes in R, when R, is small. As the Q, of the 
grid circuit is decrees^ because of a decrease in 


Section I 
Crystal Osdllotert 

R„ the effective Zl across the tube must appear 
more inductive in order for E„ to shift in the 
correct direction to compensate for the decreased 
phase rotation between grid and cathode. For this 
to occur, the net capacitive reactance of the feed- 
back arm must increase, which can only come 
about if the inductive reactance of the crystal unit 
decreases. Hence, the frequency falls, and in so 
doing, the of the crystal and grid circuit be- 
comes smaller still, so that an additional drop in 
the frequency is necessary to compensate for the 
decrease in the crysta^Q.. In the meantime, the 
bias decreases and the 'grid goes positive a larger 
fraction of the time. This tends to decrease Rp, 
which contributes even more to the drop in fre- 
quency. With all these effects adding in the same 
direction, the large frequency sensitivity of the 
Miller with changes in the grid resistance is ex- 



plate tunino capacitance - p. lit 


ffgvf 1-147. (AJ Square of oxcitation voUogo and (B) rmt valuo of output voltago of oxporlmontal Millor 
oxclllator vonut plat* tuning capacitance for variou* L/C ratio* of plat* tank. Same eryttal unit 

as wot utod for curves in figure 1-144 


WADC TR 56-156 


199 





S«ction I 

Crystal Oscillators 



sxioA SMU'sgvxnoA xndino 


WADC TR 56-156 


200 


Figure 1-748, Output and eryttal drive of experimental Miller and Pierte oseillatora as d-c plate voltage Is varied far trystal units of various 
frequencies and Pi's. The values of PI assume a rated load capacitance of 32 /i/J. Both oscillators were tuned 
to provide SO percent (max) E„ and to operate Into a 5000-ohm plate lead 



plained. The sensitivity is a maximum when is 
a maximum, for then the frequency is a minimum 
and the crystal is operating nearest its series>reso- 
nant state. This fact makes an exception to the 
rule that the larger the effective C„ the greater 
the stability. 

1-342. Of special interest in the curves of figure 
1-146 is the fact that those representing the 50- 
per-cent-maximum-output adjustment show an in- 
crease in frequency with an Increase in plate volt- 
age, whereas the curves representing a maximum 
output voltage show a decrease in frequency when 
the plate voltage is increased, even though the 
same plate voltages are applied in each case. Now, 
in the Miller oscillator, the frequency increases 
and decreases in the same direction with Rp. The 
plate characteristics of the 6C4 tube, the tube be- 
ing used when the curves in figure 1-146 were 
plotted, indicate a decrease in R,> as the plate volt- 
age increases. Thus, we should expect the change 
in frequency of the (max) E„ curves to be due to 
the change in Rp caused by the change in plate 
voltage. On the other hand, the oppositely directed 
change in frequency of the lower-percentage-Ep 
curves must be due to an oppositely directed 
change in Rp brought about by a change in the 
bias. A re-examination of the crystal voltage 
curves does indeed show at the SO-per-cent-Ep ad- 
justment that the grid excitation, and hence the 
bias, is near the maximum. In figure 1-145, it 
can be seen that for large values of grid resistance 
the changes in output voltage due to changes in the 
plate voltage cause a maximum variation in the 
crystal drive. The evidence is quite strong that 
there is an operating region between the oppositely 
changing frequency curves where the changes in 
Rp due to changes in En and E,. will annul each 
other. From an inspection of the crystal voltage 
curves in figure 1-144 we would guess that such 
operating points will lie on both sides of the maxi- 
mum-E„ region. Such a state of operation would 
be an example of “class-D” operation described in 
paragraph 1-298. 

1-343. The curves shown in figure 1-147 indicate 
the effect of variations in the L/C ratio of the plate 
tank circuit obtained by increasing the value of 
Lj in figure 1-143. Increasing L, increases the im- 
pedance into which the tube operates, and thus 
increases the r-f plate voltage. This also has the 
effect of decreasing the frequency and the react- 
ance of the crystal unit. For this reason, the 
crystal voltage does not increase in the same pro- 
portion as the plate voltage. Much greater stabil- 
ity is obtained with low L/C ratios, but much 
greater values of Ep/E„ and hence of power gain. 


Section I 
Crystal Oscillalert 

are to be obtained with large L/C ratios. Figure 
1-148 compares output-vs-drive curves for several 
different frequencies and values of PI when the 
same crystal units are used in both Miller and 
Pierce oscillators. Note that in the Pierce circuit 
the slopes of the curves consistently increase with 
an increase in PI. In the Miller circuit the tendency 
is for the slopes to increase with decreasing fre- 
quency primarily, and secondarily with the PI. 
This may be due to the fact that the gridleak re- 
sistance used in the Miller circuit was smaller than 
that in the Pierce circuit. In any event, the average 
bias will tend to be less at the lower frequencies, 
since the grid charge has more time during a cycle 
to leak off. 

MODIFICATIONS IN DESIGN OF 
MILLER OSCILLATOR 

1-344. A number of Miller oscillators currently 
being used in military equipment are illustrated in 
figures 1-149, 1-150, 1-151, 1-152, and 1-153. The 
values of the circuit parameters, where available, 
are given in the accompanying circuit-data charts. 
None of the crystal units employed in these cir- 
cuits is now recommended for equipments of new 
design. Nevertheless, all the circuits shown can be 
modified in one way'or another and used with cur- 
rently recommended crystal units which have been 
tested for parallel resonance. The necessary modi- 
fications would be those that would ensure a cor- 
rect load capacitance and would not permit a crys- 
tal to be overdriven within the expected range of 
effective resistance. The circuits illustrated sug- 
gest the wide adaptability of the Miller oscillator 
for different output requirements and uses. It is 
not possible to single out a particular circuit and 
declare this design to be preferred. The engineer 
will need to design and test his own circuit for the 
particular requirements of the equipment in which 
his oscillator is to be used. Quite often the type 
of vacuum tube or other circuit components most 
readily available influence the design. Unlike the 
Pierce, the Miller circuit must include a means of 
adjusting the plate impedance to ensure the cor- 
rect load capacitance for the crystal if the oscilla- 
tor is to operate to more than one frequency. In 
the circuits of figures 1-149 to 1-153, the switching 
arrangements of those circuits designed to operate 
over a wide frequency range are for the most part 
omitted. Most often, a separate plate coil is pro- 
vided for each crystal position. Because of space 
limitations in the circuit-data charts, occasionally 
two different components in a circuit having the 
same value or being of the same type are assigned 
the same symbol number. 


WADC TR 56-156 


201 



Saclien I 

Crystal Otcillalers 


C3 ,, 



Figure 1-149. Convantlonal Miller otcillator* using triodes 


Fig. 

Equipment 

Purpose 

F, 

Fs 

CR 

Ri 

Rj 

(A) 

Radio Modulator and Transmitter 
BC-925 

M.O. 

556- 

740 

H 

Bliley 

AR-3 

100 

2 

(B) 


mm 


II 

DC-ll-( ), 
DC-16. DC- 
26, or CR- 
1 ( )/AR 

ISO 

■ 


Target Control Transmitting 
Equipment RC«56>A 

TOT 

6585^ 

6167 

mm 


100 

100 


TMt Set TS-67/ARN-6 “ 

6.9-mc and 20.7- 
mc signals for 
testing reocivers 
in ILS 


m 

!■ 

50 


5 

Radio Set AN /ARC-1 A 


inmii 

FT 



1.6 

mniiii 


Oscillator for hetero- 
dyne freq 
meter and crystal 
calibrator 

1000 

nFi 

Dallon's 
Laborato- 
ries D-IOOO 

■1 

15 

1 


Calibration for 
vfo of signal 
generator 

5o5o 

nFi 

Holder 

FT-243 

82 

a 


I 

i 

M.O. 




51 

> 


Circuit Data for Figure 1-149. F in lie. R in kilohnu. C in ii/ii. L in fib. 


WADC TR 56-156 202 



































WADC TR 56-156 


203 









































S«ctlon I 

Ciytlal Ofcillotorf 



FIgun 1-150. ModUtd Mttimr oscMIoten uting triodm$ 


Fig. 

Equipment 

Purpose 

F, 

Fs 

F, 

Fs 

CR 

R. 

EB 

E9 



(A) 

Radio Receiver 
BC-738-DM 

Ixicel oscillator 

6688- 

6746 

F, 

3F| 

9Fi 

Holder 

FT-243 

■ 

180 

8.2 


220 

(B) 

Radio Receivers 

R-57/ARN-5 and 
R-89 ( ) /ARN-5A 


6498- 

6548 

H 




1.5 

■ 



H 

(C) 


iiiiiim 


li 

B 

|mM 


66 

12 

H 


H 

(D) 


Nautical- 
mile rao|(e- 
synehroniz- 
ing oscillator 

80.86 

H 

II 

H 

E^^Qrai^l 

■ 


8.2 


i 


Circuit Data for Figure 1-150. F in kc. R in kilohma. C in aai. L in ah. 

WADC TR 56-156 204 






































WADC TR 56-156 


205 


















































Section I 

Crystal Oscillators 




(SI 


figure I -15 1. Miller estlllator* using screen-grM tubes. (C0ectlve suppressor In beam-power tubes Is tonoeeted 
to eathodo inside tube, although external eoanettloa may be Indicated In diagram.) 


Fig. 

Equipment 

Purpose 

F, 


Ft 


CR 

Ri 

Rs 

Rs 

R4 

R, 

R« 

m 

Monitor 

ID-18/ 

CPN-2 

Mile range 
synchro- 
nizing 
(Mcillator 

98.109 

F, 



76*C oven 

270 

47 

1 

560 

0.15 

■ 

(B) 

Radio 

Trans- 

mitters 

BC-d3»- 


2000- 

4000 

■ 



Holder 

FT-164 

■ 

■ 

■ 

■ 

■ 

i 

H 

Range 

Marker 

Generator 

TD-42/ 

FPS-8 

Nauti- 
cal-mile 
range 
calibra- 
tor osc 

m 

■ 




^^1 

i 

H 




(D) 

Radio 

Trans- 

mitters 

BC-640- 

A,-B,-D 

M.O. 

6566.5- 

8666.6 





100 

0.06 

50 

85 

■ 

1 


Radio 

Trans- 

mitter 

T-171B/FR 


125- 

625 

■ 



Holder 

PT-249 

5000 

1 

l66 

100 


B 


Circuit Data (or Figure 1-151. F in fcc. R in kilohms. C in lutt. L in mI>. 


WADC TR 56-156 206 








































WAOC TR 56-156 































Mction I 

Crystal Oscillators 


TO 

AMPLIFIER 
C«6) ONLY] 


C(F) ONLY] 
AUDIO 
I 


(NOT USED > 
WITH crystal) SRI 




' ANTENNA 
hr) only] 



500 KC 
C(F) ONLY] 


TO ANTENNA 
CIRCUIT 


INCANOESCENT 
lamp, 6'Sv o.is amp 
C (F) ONLY 3 


OF USE 
IN 500‘KC 
CIRCUIT 


harmonic FILTER 


FI 

(5<S.t MC) 


T ( )/AR ^ 12 


>10 ~75 


flpctro l-ISf. Courtmwd 


Fig. 1 

Equipment 

Purpose 

(F) * 

Radio Set 
AN /CRT-3 

‘*G)bson 
Girl” 
power 06 C 



Circuit Data for Figure 1-151. F in kc. R in kilohms. C in uttf. L in ph. 

WAOC TR 56-156 208 





















































WADC TR S6-15« 






















Section I 

Crystal Oscillators 


V2 



(S) 


figure 1-752. floctron-eoup/od Milltr oscillaton. All circuits except fXi, (B), and (C) arm Irl-tmt madlUeatlonB 


Fig. 

Equipment 

Purpose 

F, 

Fr 


CR 

Ri 

Ra 

Rs 

Re 

Rs 

(A) 

Radio Re- 
ceiver and Se- 
lector BC- 
617-AZ 

Hetero- 
dyne oscil- 
lator 


F, 

4F, 

CR-1( )/AR 

56 

2.2 

2.2 

■ 


(B) 

Radio 

Transmitters 

T-3/CRN-2 

and T-3A/ 
CRN-2 

HI 

6203.7- 

6159.26 

1 

■ 

H 

CR-IA/AR 

or 

DC-17.B 

n^mi 

■ 

H 

■ 



Radio 

Transmitter 

BC-329-J 

M.O. 

m 

H 




15 

il 

■■ 

m 

(D) 

Radio Re- 
ceiver 

R-146A/ 

ARW-35 

1st het- 
erodyne 
oscillator 


H 

6Fi 

+ 

5000 



100 

4f66 



(E) 

Radio Sets 
AN/TRC-1, 

-lA, -IB, 

-1C 

Hetero- 
dyne osc 
in Radio 
Receiver 

R-19( )/ 

TRC-1 

7300- 

8750 

Wi 

NA 

CIU/U ^ 

100 

“155 

■■^55 

6 


(F) 

■1, 

Hetero- 
dyne oec 
in Radio 
Receiver 

R-19( )/ 

TRC-1 

7566- 

876« 

5Fi 

and 

6Fi 

NA. 

CR-6/W 

100 

100 

' 556 

M ' 


(G) 

Radio 

Receiver 

R-19H/ 

TRC-1 

Hetero- 
dyne oscil- 
lator 

7800- 

8750 

n 

^1^1 



100 

H 

H 



Circuit Oats lor Figure 1-152. F in kc. R in kilohms. C in mJ- L in ^h. NA; Not Applicable. 
WADC TR 56-156 210 





















































WADC TR 56-156 


211 



































Circuit Data for Figure 1-152. P in kc. R in kilohms. C in /tiJ. L in /ih. NA: Not Applicable. 

WAOC TR 56-156 212 

















































































Stdion I 

Ciytlal Ofcillcrtert 



figur* 1-153 Mlse«lfan«ew( Millmr^idllator modHItatlont 




Purpose 

ID 

Fs 

Fs 

CR 

Ri 

Rs 

Rs 

Rs 

Rs 

Rs 

Rr 

Rs 

(A) 

Galvin 

Radio 

Trans- 
mitter 
PA-8218 
P/O AN/ 
CRC-3 

M.O. for 
phase 
modula- 
tor cir- 
cuit 

937.5 

1250 

F, 

Fi 

(mod.) 

Motorola 

FMT 

1 

470 

1 

10 

47 

47 

0.1 

1 

(B) 

Radio 

Trans- 

mitter 

T-264/ 

FRC-6A 

M.O. for 
phase 
modulator 
circuit 

937 5 
1250 

F, 

F, 

(mod.) 


1 

470 

1 

10 

47 

47 

0.1 

1 


Galvin 

Radio 

Trans- 
mitter 
PA-8244 
P/0 AN/ 
CRC-3 

M.O. for 
phase 
modula- 
tor cir- 
cuit 

3750- 

5000 

F, 

F, 

(mod.) 

Holder 

FT-243 

1 

470 

1 

47 

47 

1 

0.1 

1 

(D) 

Galvin 

Radio 

Trans- 
mitter 
PA-8026 
P/0 AN/ 
VRC-2 

M.O. lor 
phase 
modula- 
tor cir- 
cuit 

3750- 

5000 

F, 

F, 

(mod.) 

Holder 

FT-248 

1 

470 

1 

47 

47 

1 

0.1 

1 


Circuit Data for Figure 1-163, F in kc. R in kilohms. C in itid. L in m 1>> unless otherwise noted. 


214 


WADC TR 56-156 



























































R» 

Cl 

Cj 

C, 

Cl 

C. i 

C, 

C7 

Cs 

C« 

Li 

u 

, 

1 

L, 

Vi 

V, V, 


5000 

25 

50 

j 

5-44 

5000 

10 

100 

100 

2000 

1 

1 

1 

7C7 

7A8 


5600 

24 

1 

6-45 

5600 

24 

100 

100 

2200 

1 

1 


7C7 

7A8 


5000 

5-44 

6000 

100 

100 

2000 

5 

2000 

5 

1 

R-P 

choke 

1 

7C7 

7A8 


5000 

5-44 

5000 

100 

100 

2000 

5 

2000 

5 

1 

R-F 

choke 

1 

7C7 

1 


WADC TR 56-156 


215 

























































Circuit Data for Figure 1-153. F in kc. R in kilohma. C in jiid. L in fik, unless otherwise noted. 


WADC TR 56-156 


216 



































































WADC TR 56-156 


217 













































Section I 

Crystal Oscillators 

Two-Tube Parallel-Resenonf Crystal Oscillators 

I-S-IS. Most of the two-tube oscillator circuits are 
designed so that the crystal is operated at or near 
its series-resonance frequency. An exception is the 
multivibrator type of circuit in which the crystal 
unit is connected between the grid and cathode of 
one of the tubes. The basic circuit is shown in fig- 
ure 1-154 (A). If there were no capacitive effects 
to consider, would be 180 degrees out of phase 
with Ek, and the 180-degree feed-back inversion 
would be accomplished entirely by Vj. In this case, 
the input impedance of V, would have to be purely 
resistive — that is, the crystal unit would operate 
at parallel resonance with the input capacitance. 
The proper phase of Eg, could also be obtained 
with the crystal unit operating near series reso- 
nance, and unless the V, input impedance were so 
reduced that oscillations could not be sustained 
under such condition there would always be the 
risk that the oscillator would jump from one equi- 
librium state to the other. In an actual circuit, the 
circuit capacitances will prevent V, from operat- 
ing into a purely resistive load, so that E,„ slightly 
lags the equivalent generator voltage — j*Eg,. The 
larger the values of R„ R,,„ Rgg, C„ and C,,, the 
nearer will the lag in E,„ approach the 90-degree 
limit. If it is assumed that Rp, is very large com- 
pared with the reactance of the Vg input capaci- 
tance, Cgz, the phase of Egj is approximately the 
same as the phase of E,„. Also, if the ratio, C,/Cgj, 
is very large, the magnitudes of E*, and Ep, are 
very nearly equal. As can be seen in figure 1-154 
(B), the lag in E,„ causes the equivalent generator 
voltage of Vg, equal to — /*Eg 2 , to lag Eg,. The cir- 
cuit will oscillate at that frequency at which the 
crystal impedance creates the necessary phase dif- 
ference between Eg, and — E*,. Note, that except 


for the possible coupling between the output cir- 
cuits of V, and V- because of the grid-to-plate ca- 
pacitance of Vg (which can be made negligible by 
the use of screen-grid tubes), the phase of E,„ 
and hence of — /lEg^, is entirely independent of im- 
pedance changes in the V- plate circuit. Thus, to 
predetermine the angle 9 in figure 1-154 (B), it is 
only necessary to consider the V, stage as a con- 
ventional vacuum-tube amplifier circuit. In turn, 
the V, stage can be treated separately as an equiv- 
alent circuit driven by a generator of voltage 
— )nEg 2 . The design must be such that when the 
crystal unit is operating at its rated X,,, the voltage 
across the crystal unit differs from the equivalent 
generator voltage, — /lEg.,, by the desired angle 9 . 
If Eg, leads E,.,, as indicated in figure 1-154 (B), 
the feed-back current through C, will be very 
nearly in phase with E,,,,. The input impedance of 
V, will appear somewhat inductive, and close to 
series resonance with C,. The smaller that Cj is 
made, the higher will be the frequency. If the plate 
circuit of V, is made inductive, E,,, (not by chang- 
ing 9 ) can be shifted to be more nearly in phase 
with Eg, : however, such operation would tend to 
become unduly critical. For example, assume that 
E „2 were rotated to where it was in phase with Eg,. 
This would mean that the phase of the feed-back 
current, Ig„ with respect to E ,,2 v.as equal to its 
phase with respect to E,.,, and this in turn would 
require that the over-all Q of Zg, and Cs in series 
be the same as the Q of Zg, alone — ^an impossibility 
unless C, is infinite. But C, cannot be made large 
without the risk that the circuit will operate as 
an RC controlled multivibrator. Thus, in the cir- 
cuit of figure 1-154, E,., must lag Eg, if oscillations 
are to be maintained. The smaller the phase dif- 
ference between Ep, and Eg,, the less will be the 
leading component of Ig, with respect to Ep,, and 


B+ B + 



WADC TR 56-156 


21S 



the less will be the leadinsr (inductive) component 
of E», with respect to I*,. Hence, the smaller the 
angle e, the more nearly must the crystal approach 
parallel resonance with the input capacitance of 
V,. Any change in the V, plate circuit that tends 
to decrease 9 therefore tends to raise the fre- 
quency. C*, can be increased by the insertion of a 
fixed capacitance to make the total approach the 
rated load capacitance of the crystal unit. 

1-346. Since the crystal unit is effectively a ca- 
pacitance at all frequencies except those near its 
points of mechanical resonance, some precaution 
must be taken in the circuit design to ensure that 
the crystal maintains control over the frequency 
and that no danger exists that the two-stage cir- 
cuit can perform as a free-running multivibrator 
with the frequency controlled by the RC constants. 
When Ca in figure 1-154 is small and is large 
by comparison, and when the plate impedances R, 
and Rb are small by comparison with the R„ of 
the tube, the feed-back voltage at the low frequen- 
cies corresponding to the RC time constants can 
quite easily be kept below the requirements for 
sustained oscillations. 

1-347. No data is available concerning the relative 
frequency stability of the parallel-resonant multi- 
vibrator type oscillator, but, from qualitative con- 
siderations only, it would seem that a performance 
equal to, and very possibly superior to, that of the 
average Pierce circuit could be expected, although 
the operation of the circuit would certainly be 
much more critical. The V, amplifier stage can be 
designed to operate into a practically purely re- 
sistive load, so that fluctuations in the V, plate 
resistance will have little or no effect upon the 
frequency. On the other hand, under these condi- 
tions 9 will be slightly negative and — will 
lead E,, unless a large reactance, or, preferably, 
a resistance, is connected in series with C,. If 9 is 
not critically small, the external circuit of V, can 
also be designed to appear as a pure resistance, 
and any variations in the plate resistance of the 
tube will have a negligible effect on frequency. 
The annulling of the stray-capacitance effects in 
the output circuits of the two tubes will require the 
use of coils, which may not be desirable if a wide 
frequency range is intended. The phase-shifting 
Qf of the feed-back circuit is computed in the 
same manner as in the Pierce and Miller cir- 
cuits except that the required phase shift is much 
smaller. 

1-348. In the design of such a circuit, the value 
of 9 can be arbitrarily predetermined. Assume that 
when R, of the crystal unit is a maximum, the 
feed-back current, I,,, is to be in phase with — /lE*,. 


Section I 
Crystal Oscillators 

With R», assumed to be large compared with the 
average PI of the crystal unit, estimate the capaci- 
tance C», required across the crystal unit for the 
Q of Z„, to equal the tangent of 9 when the X, of 
the crystal unit corresponds to the reactance of the 
rated load capacitance. Make Cj such that its re- 
actance at the operating frequency is equal to the 
reactive component of Z^, computed above. Under 
these conditions Ig, will lag — for all values 
of R.. less than maximum, since the Q of Zg, would 
be greater than tan 9 if the frequency did not in- 
crease to bring Zg, nearer its antiresonant value. 
But the increase in the frequency as a result of a 
decrease in R,, is less than that which would occur 
if the circuit were designed so that Ig, woulc' lead 
— ^lEg- for most values of R,.. Also, if the feed-back 
circuit is designed for series resonance when R, 
is a maximum, the values of C-, should prove more 
practical, the operation will be less critical, and 
there is the assurance that all values of R,. will 
permit oscillation. The plate-circuit impedances of 
the two tubes are next determined so that the 
feed-back voltage is sufficient to maintain oscilla- 
tions when R, of the crystal unit is a maximum. 
Generally, neither tube should operate into a load 
exceeding 5000 ohms. The plate voltages are 
chosen so that the crystal unit cannot be driven 
beyond the rated drive level. With the phase char- 
acteristics of the circuit determined, more or less 
by design, to ensure a proper load capacitance for 
the crystal, it may be that the optimum operating 
voltage will be more readily determined through 
experiment. At equilibrium, the total voltage gain 
of the loop, from Eg, to E,„ to Eg^ to E,,. and back 
to Eg,, must be equal to unity. Thus, 


G, G2 Ga G, 


E,i 2 Eg 2 

Eg. ■ Ep, 


Ep2 

Eg2 


Eg. _ j 

Ep2 

1—348 (1) 


where, referring to the circuit in figure 1—154, 

G. = Ep,/E„=^ =^g„.Rg 

Kpi -T ^pi 

1—348 (2) 

Gj = Egj/Ep, = 1 ' 1-348 (3) 

Gs = Ep2/Eg2 = ^ >-1 = g:n2 Rb 

Rp2 "T ^p2 

1—348 (4) 

Gg = Eg,/Ep2 = 1 1-348 (5) 

So 

G. G 2 Gs G 4 = gn,l gm 2 R» Rb = 1 

1—348 (6) 


WADC TR 56-156 


219 



Section I 

Crystal Oscillators 


Equation (6) is only a first-order approximation 
in which it is assumed the plate resistances of the 
tubes are very large compared with the external 
plate impedances, Z„, and which, in turn, are 
approximately equal to R, and Rb, respectively. 
Also, it is assumed that X,., is small compared with 
Zfi, and that either X,., is small compared with 
Zj, or the tendency towards a series-resonant rise 
in voltage across Zg, is sufficient to make equation 
(5) approximately correct. 

1-349. The output is most often taken from across 
a 500- to 1000-ohm resistance between the cathode 
of Vi and ground. In this event, R.^ connects di- 
rectly to the cathode of Vj — not to ground. The 
cathode output is quite useful for matching to low- 
impedance inputs, such as would occur, for ex- 
ample, when feeding a coaxial line. Regardless of 
where the output is obtained, it can be seen that 
its amplitude cannot be expected to greatly exceed 
that of the Pierce circuit. Since two amplifier 
stages are required and no additional gain is pro- 


duced, there can be little advantage in using the 
multivibrator circuit unless thermostatic control 
of the temperature is employed and the design is 
such as to ensure greater frequency stability than 
can be achieved in the Pierce circuit. The use of 
a single dual-type tube offers greatest economy. 
The principal advantages of the circuit are its 
relative independence of fluctuations in the tube 
voltage, and its adaptability for impedance-match- 
ing to low-impedance output circuits, 

1-350. Figure 1-155 illustrates three multivibra- 
tor-type crystal oscillators that were designed 
for use in Diversity Receiving Equipment AN/ 
FRR-3 ( ). Circuit (A) is a later-model replace- 

ment of circuit (B). Very possibly the preference 
for (A) is at least partly due to a desire to elimi- 
nate the variable effects of the inductors in (B) 
with changes in frequency. From the data avail- 
able it cannot be said that the crystal in (B) is 
not actually operating at or very near its series- 
resonance frequency. The state of operation of the 


R4 



figure 1-155. 7wo-$tag» paralM-n$onant o*elMaton of muhivibrator typ*. 


Fig. 

Equipment 

Purpose 

Pi 

CR 

Ri 

Rt 

R> 

R4 

Rs 

R« 

Rt 


(A) 

Diversity 

Receiving 

Equipment 

AN/FRR-3A 

Local 

oscillator 

1400- 

3800 

Holder 

FT-249 

250 

100 

5 

■ 

0.6 

0.16 


■ 

(B) 

Diversity 

Receiving 

Equipment 

AN/FRR-3 

Local 

08C. 

1400- 

3800 

Holder 
FT-249 
(Entire 
circuit in 
65* C 
oven) 

250 

i 

■ 

■ 

1 

■ 

■ 

■ 

1 

Diversity 

Receiving 

Equipment 

AN/FRR-3A 

6f5 

with 

AFC 

react- 

ance 

tube, 

V. 

462.45 


50 

mii^ 

10 

mill 

■ 

I 

6 

1 


Circuit Data for Figure 1-155. F in kc. R in kilohma. C in ii/it. L in ^h. 


WADC TR 56-156 220 


































SccKen I 
Cryitol OtciilcrtArt 


R5 



flgur* I- 155. Continued 


R« 

C, 

C: 

Cj 

C, 

Cs 

C. 

Ct 

Cs 

L, 


V 1 -V 2 

Vj 


80 

10 

100 

2000 







6SN7GT 

6AC7 


10 

200 

75 

50 

miiiiii 

100 

50,000 

100 


■ 

m 

■ 

0.6 


100,000 

■ 

250 

50 

10 

100 

2000 


2200 

6SN7GT 

6V6GT 


WADC TR 56-156 


221 







































SacHon I 

Cryttql OKiilaton 

crystal will vary quite widely with changes in the 
tuning of the LjCa tank. Circuit (C) is a beat- 
frequency oscillator which is crystal-stabilized 
when switch S, is in the crystal position, as shown. 
The triode-connected beam power tube serves as 
a reactance tube, which effectively shunts the 
crystal with a capacitance that varies with the 
bias supplied to the control grid of the tube. The 
bias, in turn, is controlled by the a-f-c discrimi- 
nator circuit in a teletype terminal. The purpose 
of the circuit is to ensure that the beat frequency 
remains constant even though the frequency of the 
incoming signal should vary slightly. If the beat 
frequency tends to drift, the sign and magnitude 
of the discriminator output causes the bias of the 
reactance tube to effectively change the load ca- 
pacitance of the crystal unit in such a direction 
that the frequency of the oscillator rises or falls 
by approximately the same number of cycles per 
second as does the incoming signal. 

Oscillators with Crystals Having 
Two Sots of Eloctredos 

1-351. The original crystal oscillator devised by 
Dr. Nicolson, as well as a number of the earlier 
crystal oscillators tested by Dr. Cady, employed 
crystals with, effectively, two pairs of electrodes. 
The basic circuit is shown in figure 1-156. The re- 
quired phase inversion of the amplifier output 
voltage is provided by the crystal unit operating 
at a mode for which the polarities of the plate 
and grid terminals with respect to ground are 180 
degrees out of phase. The circuit shown operates 
the crystal unit very near its series-resonance fre- 
quency. In practice, a capacitor is normally con- 
nected between crystal and ground, so that the 
circuit is more commonly employed for parallel- 
mode tested crystal units. Still, it is not without 
some license that we classify this type of oscillator 
as a parallel-mode type. The crystals most appli- 
cable for this class of circuit are the very-low- 
frequency elements of the X group, which vibrate 
in lengthwise extensional or flexural modes. The 
electrode connections that permit the desired 
phase inversion depend upon the particular crystal 
element. Assume that electrodes numbers 1 and 3 
are on one side of the crystal, and that 2 and 4 
are on the opposite side, as indicated in figure 
1-156 (A). For a flexure element, such as element 
N, where electrodes 1 and 3 parallel each other 
down the length of the crystal, as shown in figure 
1-156 (B), the flexure mode is excited when the 
potential across 1 and 2 is oppositely polarized to 
that across 3 and 4. If the same electrode arrange- 
ment is to be used to excite an extentional mode 


(or the flexural mode of the duplex element J) 
the polarities of the two sets of electrodes must 
be in phase. In this case, the connections of one 
set of electrodes should be reversed in the circuit 
shown in figure 1-156 (A). For example, plates 2 
and 3 should be connected to ground and plate 4 
should be connected to the grid, if the proper phase 
inversion is to be obtained. A crystal having the 
two sets of electrodes at opposite ends of the crys- 
tal, as shown in figure 1-156 (C), would be driven 
at the second harmonic of the length extensional 
mode (or of the flexural mode of a duplex crystal) , 
if connected as shown in figure 1-156 (A). Greater 
stability and a smaller crystal arc possible for a 
given frequency by operating at the fundamental 
mode. To permit this, if the crystal unit is plated 
as shown in figure 1-156 (C), the connections of 
one pair of electrodes should be the reverse of those 
shown in figrure 1-156 (A). If it can be assumed 
that the current in the grid circuit is negligible 
compared with the crystal current between ter- 
minals 1 and 2, and if the stray capacitance be- 
tween the two sets of electrodes is ignored, the 
equivalent circuit between terminals 1 and 2 
will appear approximately as shown in figure 
1-156 (D). L, C, and Co represent the parameters 
of a fully plated crystal. A more exact analysis of 
this type of crystal unit can be found in the book 
“Electromechanical Transducers and Wave Fil- 
ters” by W. P. Mason, D. Van Nostrand Co. 

1-352. Figure 1-157 shows a practical oscillator 
design employing crystal units having two sets of 
electrodes. Although the electrode connections 
shown for CR would indicate that the plate and 



2 

<81 (C) (0) 

figure 1-156. Basic circuit ot oseillator using crystal 
with two pairs of elaetrodos 


WADC TR 56-156 


222 




SacHon i 
Cryctal Oicillatort 



flgun t-IS7. Practical crystal-oscillator dmsign omploying vory-low-frsquoitcy 
crystal unit -with two sols of oloctrodos 


grid terminals always connect to the same side of 
the crystal unit, the actual connections will depend 
upon the particular element used. The variable 
capacitor Ci permits a frequency adjustment of 
approximately 60 parts per million. To ensure that 
the crystal unit is operating into its rated load 
capacitance, the exact frequency of a test crystal 
should be known when it is at series resonance 
with its rated capacitance. To a first approxima- 
tion, the terminal that is to be connected to the 
grid can be assumed to be open-circuited, so that 
the resonance to be tested is that between the rated 
Cl and one half of the crystal. With the test crystal 
connected in the oscillator circuit, C, can be ad- 
justed to provide an output at the previously 
measured “rated” frequency of the test crystal. 
This adjustment therefore will provide the rated 
load capacitance for all crystal units of the same 
type. The varistor is inserted to protect the crystal 
unit from overdrive, and to ensure a stable output 
voltage. As recommended by Bell Telephone Lab- 
oratory engineers, the nominal values of R„ C,, 
C„ and C .1 that provide satisfactory operation in 
the 1 . 2 - to 10 -kc frequency range are given in the 
following table. The values shown will provide a 
direct current in M of approximately 12 micro- 
amperes. 


Frequency 

Range 

(kc) 

R. 

(kil- 

ohms) 

c. 

(w^f) 

c, 

(mt) 

c, 

(M/if) 

1.2 — 1.5 

100 




1.5 — 2.0 

100 



Hia 

2.0 — 2.5 

100 

160 


500 

2.6 — 3.2 

100 

160 


600 

3.2 -- 4.5 

100 

120 


500 


Frequency 

Range 

(kc) 

R. 

(kil- 

ohms) 

c, 

(mm^) 

c, 

(mm^) 

C 3 

(/*Mf) 

4.5 — 6.7 

100 

120 

700 

250 

6.7 — 8.0 

100 

90 

500 

250 

8.0 — 10.0 

51 

90 

1000 

0 


1-353. Figure 1-158 shows the crystal oscillator 
in Test Set TS-251/UP, which employs a duplex 
crystal element. The crystal circuit is used to syn- 
chronize the blocking oscillator at a frequency of 
1818.18 cps. The output of the blocking oscillator 
is for counting down to 303.03 pulses per second, 
which, in turn, are used in checking Loran pulse- 
repetition rates. A CR-ll/U crystal unit is used 
which has a resonant frequency of 1817.44 ±0.3 
cps at 75° Fahrenheit. The rated maximum effec- 
tive resistance of the crystal is 30,000 ohms, and 
its rated maximum permissible current is 0.03 
milliampere. The fixed capacitance paralleling the 
variable capacitance is used only if necessary. The 
varistor is rated at 1 ma/14V for temperatures 
between 75 and 86 degrees Fahrenheit. 

Crystal and Ma9ic-Ey« Rasenaace Indicator 

1-354. An interesting application of a parallel- 
resonant crystal circuit is the tuning indicator 
shown in figure 1-159. When the tuned frequency, 
F„ of a variable oscillator is equal to the antireso- 
nant frequency of the crystal unit in parallel with 
the input capacitance of V,, a magic-eye tube, the 
excitation of V, is a maximum, as is the current 
through R,, and, hence, also the shadow angle of 
the indicator. The circuit thus provides a con- 
stant visual crystal check on the tuned oscillator 
frequency. Different crystal units can be switched 
in for different channels. 


WADC TR 56-156 


223 













Sactlen I 

Crystal Oscillators 



Figure 1-158. Oup/ex-ofoctrodo irystal eireult in Tost Sot T5-251/UP for synchronizing blocking oKlIlator 
at I SI 8.18 cps. Resistanco is given in kllohms, capacitance in micromicrofaradt 



Figure 1-159. Crystal and magic-eye resonance indicator 


Fig. 

Equipment 

F, 

CR . 

R, 

Ra 

(A) 

Radio Transmitter BC-696-A 

3000- 

DC-8-C.-, 

5 1 

0 39 


4000 

D,-K 



(B) 

Radio Transmitter BC-457-A 

4000- 

DC-8-C,- 

10 

0.39 


5300 

D,-K 



(C) 

Radio Transmitter BC-458-A 

5300- 

DC-8-C. 

15 

0,39 


7000 

D,-K 



(D) 

Radio Transmitter BC-45d-A 

7000- 

DC-8-C. 

5.1 

0.39 


9100 

D,-K 




Circuit Data for Figure 1-159. F in kc. R in kilohms. C in mmI- 


224 


WADC TR 56-156 



























S*cH»n I 
Crystal Oscillators 


SHIES-RBSONANT CRYSTAL OSCILLATORS 

1-355. For maximum frequency stability it is gen- 
erally preferable to operate a crystal unit at its 
series-resonance frequency, but series-mode cir- 
cuits are most widely used for overtone operation. 
At series resonance the crystal element appears as 
a resistance, so that in the normal circuit it can be 
short-circuited or replaced by a comparable resist- 
ance without stopping oscillations. Series-resonant 
oscillators generally have smaller outputs than do 
oscillators of the parallel-resonant type. Also, 
series-resonant oscillators usually require more 
circuit components, and hence are not often used 
except in the very-high-frequency range. In gen- 
eral, the operation of the series-mode circuits is 
less complicated than that of the parallel-mode 
oscillators. Nevertheless, the circuit design be- 
comes incrjeasingly critical at the higher frequen- 
cies and higher overtones. The stray capacitances 
must be kept tc a minimum, and all leads must be 
as short as possible. It may be necessary to nullify 
the crystal shunt capacitance, Co, by connecting 
across the crystal unit an inductor that is anti- 
resonant with C„ at the operating frequency. It 
may also be desirable to connect a capacitor in 
series with the crystal unit, to tune out the stray 
inductance of the crystal leads. Tuned circuits 
must be provided if a crystal unit is to be driven 
at a particular overtone mode. Quite often, satis- 
factory operation is obtained simply by designing 
a conventional variable-tuned oscillator to operate 
at the desired frequency, and then inserting the 
crystal unit in a plate tank or feed-back circuit. 
There will be a range of tuning adjustments in 


which the crystal can assume control and hold the 
frequency very nearly constant. As the tuning ad- 
justments are varied beyond this range, the con- 
trol becomes quite unstable or ceases altogether. 
Usually, the region of stable control becomes 
smaller as the overtone order is increased. If 
broad-band operation is desired with no tuning ad- 
justment other than the selector switch for chang- 
ing the crystal, additional precautions must be 
taken to ensure that oscillations cannot be main- 
tained except when the crystal impedance is small 
— that is, the crystal unit is operating near series 
resonance. For maximum frequency stability, the 
effective resistance of the circuit facing the crys- 
tal unit should be as small as possible. At the 
higher frequencies, the stray capacitances limit 
the impedances obtainable from the tuned circuits, 
thereby making them more selective and hence 
more effective in influencing the frequency and in 
increasing the instability. 

1-356. The series-mode oscillators most widely 
recommended are listed in the following table, and 
rated according to their relative design and per- 
formance characteristics. A rating of 1 represents 
the top relative superiority in the corresponding 
characteristic. It should be understood that the 
ratings are based upon average qualitative results 
which might well be contradicted by the data of 
individual investigators. Any one of the series- 
mode circuits expertly designed could surpass the 
performance of a poorly designed circuit rated 
higher in a particular characteristic. The fre- 
quency-stability rating assumes average oven and 
voltage regulation. 


Rs 

R4 

Rs 

Rs 

C, 

V, 

1.5 

1000 

51 

0.02 

50,000 

1629 

1.0 

1000 

51 

0.02 

50,000 

1629 

1.0 

1000 

51 

0.02 

50,000 

1629 

1.5 

1000 

51 

0.02 

50,000 

1629 


WADC TR 56-156 


225 





























Section I 

Crystal Oscillators 


Symbol of Oscillator 

A 

B 

C 

D 

E 

F 

G 

H 

I 

B 

K 

Bi 

Frequency Stability (%) 

0.0001 

p 

b 

0 

to 

0.0005 



0.0004 

0.0004 

0.0002 

0.0015 

0.002 

0.002 

0.0015 

Power Output 

6 

5 

3 

1 

3 

5 

4 

4 

4 

2 

1 

3 

Versatility 

4 

4 

2 

1 

2 

3 

2 

3 

8 

3 

2 

3 

Upper Frequency Level 

5 

1 

3 

2 

2 

2 

2 

3 

3 

3 

4 

3 

High-Resistance Crystals 

2 

6 

3 

4 

4 

4 

2 

4 

4 

4 

1 

3 

Ease of Adjustment 

6 

7 

1 

3 

2 

2 

2 

5 

5 

5 

4 

5 

Untuned Bandwidth 

5 

5 

2 

2 

1 

2 

1 

6 

6 

6 

4 

6 

Frequency Multiplication 

5 

5 

2 

4 

4 

1 

5 

5 

3 

3 

6 

6 

Low Harmonic Output 

1 

2 

2 

5 

6 

5 

5 

4 

4 

4 

3 

2 

Circuit Simplicity 

4 

5 

3 

2 

a 

2 

2 

3 

3 

3 

4 

4 

Isolation from Load 

1 

2 

3 

4 

B 

2 

4 

4 

4 

4 

4 

4 

Low-Frequency Operation 

1 

6 

3 

6 

5 

5 

2 

4 

4 

4 

1 

2 


The oscillator symbols in the foregoing table cor- 
respond to the respective index letters of the 
oscillators listed below. 

Names of Series-Mode Oscillators 

A. Meacham Bridge 

B. Capacitance Bridge 

C. Butler, or Cathode-Coupled 

D. Grounded-Cathode, Transformer-Coupled 

Type 

E. Grounded-Grid, Transformer-Coupled Type 

F. Grounded-Plate, Transformer-Coupled Type 

G. Transitron 

H. Impedance-Inverting Transitron 

I. Impedance-Inverting Pierce 

J. Impedance-Inverting Miller 

K. Grounded-Cathode Two-Stage Feedback 

L. Modified Colpitts, C.I. Meter Type 

Meacham Bridge Oscillator 

1-357. The Meacham bridge oscillator, illustrated 
in figure 1-160, provides the greatest frequency 
stability of any vacuum-tube oscillator yet de- 
vised, but the region of maximum frequency sta- 
bility is limited to the lower frequencies because 
of the increased effect of the stray circuit capaci- 
tances when the frequency becomes greater than 
a few hundred kilocycles per second. The oscillator 
is of the crystal-stabilized type employing tuned 
circuits. At frequencies above 1000 kc the effect 
of the stray capacitance is sufficient to reduce the 
stability to a point where little is to be gained 
by the use of the Meacham circuit. The oscillator is 
principally employed with GT-cut crystals in fre- 
quency standards, to generate frequencies of 100 
kc. In figure 1-160, it can be seen that if the bridge 


were perfectly balanced there would be no excita- 
tion voltage. At the start of oscillations, the ratio 
of Ri (practically equal to the series-arm R of the 
crystal unit) to R, is smaller than the ratio of 
the Rj to Ri. But R4 is a thermistor — it is the 
resistance of a tungsten lamp which sharply 
increases in value as the temperature rises. (A 
semiconductor such as carbon, silicon, or germa- 
nium can be used, in which case the resistance will 
decrease with temperature. The negative tempera- 
ture coefficient of the semiconductor is generally 
larger than the positive coefficient of tungsten, but 
the semiconductor thermistor is more expensive 
and is much more difficult to duplicate because of 
its great sensitivity to impurities.) As oscillations 
build up, the current through Rj increases to a 
point where the heat generated from the power 
losses raises the the thermistor temperature, and 
hence the resistance, to a point where the bridge is 
almost balanced. Equilibrium is reached when the 
imbalance of the bridge is just sufficient to sup- 
ply heat to the thermistor at the same rate at 
which it escapes. For maximum amplitude stabil- 
ity, the ambient temperature of Ri should not be 
permitted to vary over a wide range. Normally, the 
tungsten lamp will heat to a dull red of approxi- 
mately 600 degrees centigrade. A variation of over 
100 degrees in the ambient temperature could have 
a significant effect on tlie equilibrium power losses 
in the thermistor, if extreme precision were de- 
sired. The operating temperature of the lamp is 
very low compared with the rated temperature, 
and consequently the lamp can be expected to last 
indefinitely. The oscillator should be designed and 
adjusted so that the phase shift occurs entirely 
in the bridge. That is, the tube should operate into 


WADC TR 56-156 


226 






















SecHon I 
Crystal Oscillators 



Hgun 1-160. Ba*k cirtuit of Moaeham bridge-stabilized oscillator 


a pure resistance, so that the instant the plate 
current is maximum the peak transformer volt- 
ages should occur with polarities as indicated in 
figure 1-160. Transformers having powdered-iron, 
toroidal cores can provide a coefficient of coupling 
very close to unity in the low-frequency range. 
The following analysis of the frequency stability 
and the activity stability of the Meacham oscilla- 
tor, except for minor deviations and extensions, 
has been guided by the postulates and basic con- 
siderations as presented by W. A. Edson.* 

FREQUENCY STABILITY OF MEACHAM 
BRIDGE OSCILLATOR 

1-358. First, we shall assume that the vacuum 
tube in figure 1-160 operates into a purely resistive 
load, and that the entire phase reversal takes place 
in the bridge transformer. The phasor diagram in 
figure 1-161 (A) shows the relation of the volt- 
age EIo to the other voltages of the bridge network. 
(Refer to figure 1-160 for voltage symbols.) Next, 
assume that some change in the capacitance of the 
circuit requires that Eo be shifted in phase by a 
very small angle equal to but that the change 
is so small that the magnitude of all the bridge 
voltages can be assumed to remain constant. In 
order to produce the phase shift it can be seen 
that E„ and hence the current through R„ must 
be rotated by an angle e. In a triangle with angles 
A and B opposite to sides a and b, respectively, 

* Vacuum Tu6e Oscillators, John Wiley and Sons, 1953. 

WADC TR 56-156 


o ][^ 

— : — T- = ^ (Law of Sines). Likewise, in the 

sin A sin B 

triangle EoEjE,, 

Eo/sin d = E 4 /sin <t> 
or 

sin 6 = sin 0 1 — 358 (1) 

E4 

Since we are assuming that both d and 6 are very 
small, equation (1) can be written, approximately, 

e 1—358 (2) 

E4 



(B) 

Figure 1-161. Phasor diagrams of bridge voltages in 
Meacham oscillator. Angles 6 9 (greatly magni- 

fied) represent small shifts in phase when crystal unit 
is operating slightly off series resonance 


227 



Section I 

Crystal OKillatert 

The current through the input transformer can 
be assumed to be negligible compared with the 
total current through the crystal, so the current 
through Rj is essentially the same as that through 
the crystal. Under these conditions, as is shown 
in paragraph 1-241, 


6 ~ tan d 


Xe 

Ri + R2 


2LAu 
R -|- R 2 


1—358 (3) 


where L and R are series-arm parameters of the 
crystal unit. By rearranging equation (3) and di- 
viding by <u, the fractional change in frequency 
required to produce an angle B is found to be 


A« 0 (R -h R 2 ) e R„VC , ,,, 

IT = 2a,L = -z7L~ 

where Rc = R -f R, is the total resistance of the 
crystal side of the bridge, and C is the series-arm 
capacitance. On substitution of equation (2) in 
equation (4), it is found that 


Ao? 0 Eo Rc / C 
o> 2E4 \ L 


It now remains to determine the magnitude of 
for a small change in the capacitance of the cir- 
cuit. The most likely changes in capacitance take 
place in the grid circuit, the average aC* usually 
being on the order of 10 times the average ACp in 
the plate circuit. Looking away from the grid, it 
can be seen that when the bridge is very nearly 
balanced, the grid faces a resistive impedance 


N/ (Ri -f- R3) (R2 + R4) 

Ri "f" R2 "i" R3 "I" R4 


1—358 (9) 


The capacitance Cg will have been adjusted to be 
effectively antiresonant with the leakage induct- 
ance of the transformer, which inductance can be 
imagined to be in parallel with R*. Still looking 
away from the grid, we can imagine a generator 
connected between grid and cathode. If the capaci- 
tance should change by a small amount aC», the 
ratio of the excess reactive component of current 
to the resistive component becomes R,<oACg. This 
will equal the tangent of the phase shift, <i>, which 
is sufficiently small for tan to be assumed to 
equal ip. Thus, with 4 , = R,<uAC(, on substituting 
equation (9) for Rg, equation (8) becomes 


Equation (5) indicates that the more nearly bal- 
anced the bridge (the smaller the Ep/E* ratio), 
the greater will be the frequency stability. 

Now, 

^ 

Letting (R3 -f- R4)/R4 = (Ri + R^l/Ra = m. we 

see that E0/E4 = • 

E, 

Also, since 


Aw _ / R2 Ra — Ri R4 \ (Ri Ra) (R2 ~1~ R4) 
<0 \ R2 R4 / (Ri -j- Ra + Ra + R4) 


Re N,^ ACg 

2L 


1—358 (10) 


Now, let us assume that Ri and Rg remain fixed. 
As Ri is varied, R, must vary in direct proportion 
to keep the bridge balanced. If R, = kR„ R4 will 
always approximately equal kR,. Substituting 
these values of R, and R« in all terms where the 
error introduced can be considered negligible, equa- 
tion (10) becomes 


E = E 51—^ 

'^•VR8 + R4 R. + R2y 


^ Ri* (kRa - R4) (1 4- k) N,* AC, 

0) kRa 2L 

1-358 (11) 


where Rp = Ra + R4, then 


If R4 is expressed as being equal to R, (k — i), 
equation (11) becomes 


E0/E4 = m(^-|i) = Ra/R4 - Ri/Ra 

On substitution in equation (5), we have 


Aw 

w 


/ Ra R3 — Ri R4 \ Re ^ 

I, RaR4 /2VI7C 


1—858 (8) 


^ ^ (1 -I- k) i Ri'* Ng’* ACg 
w “ 2kL 


1—858 (12) 


1-869. Equation 1 — 358 (12) does not quite indi- 
cate the relations among all the circuit parameters 
that are effective in providing an optimum fre- 


WADC TR 56-156 


228 



quency stability. It is first necessary to determine 
how i, which is a measure of the imbalance in the 
circuit, is dependent upon the other parameters. 
For this purpose, it is necessary to find that value 
of i which must exist in order for the feed-back 
voltage to be at equilibrium. It will be assumed 
that the r-f plate current. Ip, is equal to gmEg. If 
this assumption is not warranted, g„ in any of the 
following equations can be replaced by /i/ (Rp -|- Zp) . 
To a first approximation, 

Zp = Np* R. Rt/(Rc + Rt) 1—359 (1) 


and 


El — Ep/Np — Ip Zp/Np — g„ Eg Np Rp Rx/ (Ro + Rt) 

1—369 (2) 

also, Eg = NgEp, and E„ is given by equation 
1 — 358(7). On substitution in equation (2) 


1 = go. Np Ng .^Rc Ra - Ri Rt)/(R. + Rt) 

1—359 (3) 

Equation (3) is the equilibrium feed-back equa- 
tion for the Meacham oscillator. On expressing Ra 
and Ri as functions of R, and R^, it is found that 
at equilibrium 


(1 + k) Rp 

gn. Np Ng Ri Rj 


1—359 (4) 


Substituting (4) in equation 1 — 358 (12) 


^ = (1 + k)^ Ri Rp Ng AC, 
« 2 k g,p Rj Np L 


1—359 (5) 


Section I 
Crystal OsciUaters 

Also, caution must be taken that in improving the 
stability in one respect, it is not impaired to a 
greater extent in another. Since the expected ACg 
is approximately 10 times the expected ACp, the 
ratio of the right-hand sides of equations (5) and 

(6) can be equated to 1, with 10 aCp substituted 
for aCg. On thus dividing (5) by (6) 

^ 10 (1 + k)^ Ri Ra 

k Rp' Np' 

or 

.t 2 ,xt 2 10(l-|-k)'R. 10 (1 + k)^ (m - 1) 

" kmRp "■ km' 

1—359 (7) 

With the oscillator designed according to equation 

(7) , average capacitance variations in the plate 
and grid circuits will have approximately equal 
effects upon the frequency. When the square root 
of equation (7) is combined with equations (5) 
and (6), and Rp is expressed as mR,/(m — 1), 


Aw _ ACg m' Ri j 1 -|- k 
w 2 L g,„ V 10 k (m — 1) 


1—359 (8) 


and 


^ / lO (1 + k) 

w 2Lg^ \ k (m - 1) 


1—359 (9) 


If the oscillator is to be designed on the basis of 
equation (7), it can be seen that k should be made 
as large as is practicable, and m should be such 
that the factor mVVn^ — 1 is a minimum. It can 
be shown that this occurs when 


In a similar manner, in equating to ZpuACp, it 
can be shown that for small changes in the plate 
capacitance 


^ = R.^ Np ACp 

o) 2 (1 + k) g,p R/ Ng L 


1—359 (6) 


It can be seen that greater stability is to be had 
when gn, is a maximum and when the ratio m = 
R«/Rt is small. For changes in Cg, the optimum 
value of k is 1 (when (1 -f k)Vk passes through 
a minimum) . For changes in Cp, it would be desir- 
able to have k as large as practicable. A further 
consideration is to so proportion the parameters 
that the expected variations in C, and Cp will have 
the maximum opportunity to cancel in their effects. 


m = 4/3 1—359 (10) 

Frequency stability, of course, is not the only con- 
sideration; there are also the vacuum-tube and 
thermistor characteristics and the power rating of 
the crystal unit that must be taken into account 
in deciding upon the optimum parameter rela- 
tions. Remember, that in equations (4), (5), and 
(6), gm can be replaced by the more exact term 
/i/(Rp -f Zp). Certainly, a high-mu tube is to be 
preferred, and when it is operated class A the sec- 
ond-harmonic output can be expected to be at least 
65 db below the fundamental. The screen voltage 
should be fairly high, in order to increase gm. Nor- 
mal operating voltages can be employed, but Eg 
should not be allowed to drive the grid positive. 


WADC TR 56-156 


229 



Saction I 

Crystal Oscillators 

ACTIVITY STABILITY OF MEACHAM 
BRIDGE OSCILLATOR 

1-360. Starting with Eo, the input to the grid 


transformer, we see that at equilibrium the prod- 
uct of the gains of all the stages, from Eo back to 
Eo, must be equal to 1. Thus, 


Gain: 


Voltage: 


Gi G2 Ga G 


t — 


Eo 


G, 


N. X X 1/N, X 

i ^ i ^ i 

= E,/Eo^ g Gz = Ep/E,^ j, Ga = E,/Ep^ ^ 



1 

E. 


From equation 1 — 358(12) it can be seen that in 
the interest of frequency stability, i, and hence the 
imbalance of the bridge should be as small as pos- 
sible. Fortunately, this condition also agrees with 
the requirements of high activity stability, for the 
smaller the difference of the actual thermistor re- 
sistance from a value equal to kR:, the larger will 
be the percentage change in that difference for a 
small change in the thermistor voltage. Equation 
1 — 358(7) can be written 

Eo = G 4 E. = i R. R2 E,/k Ro" 1^60 (1) 

where i = (kRj — R4)/R2 and G4 is the gain of 
the stage. In the over-all gain equation, above, Gj 
and Gj can be considered constant, so that G« 
primarily has the function of compensating any 
changes in G2 of the vacuum tube. From equation 
(1), it can be seen that iE,/Eo can be considered 
a constant. Or, in the over-all gain equation we 
see that 

Gi G2 G3 i R, R2/k Re* = 1 1—360 (2) 

or that 


If the effects on E, due to the changes in E. and i 
exactly cancel so that dEo = 0, then, by equations 
(3) and (5) 

— dG2/G2 “ di/i = — dEe/Eg 

Under these circumstances it can be seen that the 
percentage change in the activity is exactly equal 
to the change in the gain of the tube. If the ther- 
mistor is to be effective in preventing the ampli- 
tude of the output from changing significantly 
with changes in G2, clearly an increase in E, must 
produce a decrease in E„. We can define the activ- 
ity sensitivity of the bridge to be 

s E, di/i dE, 1—360 (6) 

The sensitivity is thus defined as the percentage 
variation in i per percentage variation in the volt- 
age across the bridge. The problem now is to con- 
vert equation (6) into a function (equation 14) of 
the circuit constants so that s can be predeter- 
mined by the design engineer. From equations (5) 
and (6) we find that the percentage change in E„ 
per percentage change in E, is 


G2 i = k Rc* Np/Ri R2 Ng = constant 
On differentiating, 

i dG2 -h G2 di = 0 


di dG 2 

i G 2 


1—360 (3) 


Equation (3) shows that for a given percentage 
change in the gain of the tube, the smaller the 
value of i, the smaller need be the change in R, to 
restore equilibrium. Since G2 ^ gmZ,„ we can write 


di/i = - dg„/g,„ 1-360 (4) 

On differentiating iE,/Eo = C, where C is a con- 
stant, we find that 


i dE, -f E. di - C dEo = 0 1—360 (5) 


E, dEo 
E„ dE. 


s — 1 


1—360 (7) 


In practice, E„/E, (^ G,) can be on the order of 
0.003 or smaller; so, if the change in Eo is com- 
parable to that of E, in magnitude, excellent am- 
plitude stability will be achieved. The stability 
depends first upon the magnitude of i, and sec- 
ondly, upon the sensitivity of the thermistor. The 
latter is defined as 


E 4 dR^ _ d(log R 4 ) 
R4 dE4 d(iog E4) 


1—360 ( 8 ) 


Figure 1-162 shows the resistance-voltage char- 
acteristics of a number of tungsten lamps for 
ambient temperatures at room values. For lower 
ambient temperatures, the curves would be shifted 
to the right somewhat, and for higher tempera- 
tures, to the left. Since the curves are plotted on 
log paper, according to equation (8) it can be seen 


WADC TR 56-156 


230 




0.03 0.10 0.3 1.0 3 10 30 

APPLiEO POTENTIAL -VOLTS 

FIgun Ui62. katiatancm of typical fungatan lamps 
vnut appllad veltaga and pawar dtsalpatlon whan 
tha amblant tamparatvn la 300° Kalvin 
acala (approxtmatmly 27° C) 


that the thennistor sensitivity S at a griven value 
of E 4 is the actual slope of a curve at that point. 
It is important, of course, to operate the thermis- 
tor at a voltage where the slope approaches a maxi- 
mum. It is convenient to express the bridge s as 
a function of the thermistor S. 

O' ■ ^ 4L 

Since 1 = 5 , then 

Its 


di = 


or 


dR 4 _ k dR 4 

R 2 R 4 


1---360 (9) 


E.di ^ k E. dR 4 ^ kS / E.dE 4 \ 
i dE, i R 4 dE, i \ E 4 dE, / 

1-360 (10) 


Now, E4 = E2 — Eo = Ea, but dE4 ( = dE* — dEo) 
is not approximately equal to dEa. So dE4/E4 == 
(dEa — dEo)/Ea = (dE, — mdEo)/E„ where m = 
R E 

^ On substitution for dE 4 /E 4 in equation 

Xv2 ^2 
( 10 ), we find 


rearranging, we have 


Section I 
Crystal Oscillators 


kSE, - mkSEo 
iE, — mkSE„ 


1—360 (12) 


The term mkSE„ in the numerator can be consid- 
ered negligible, and dropped. After expressing E, 
in terms of equation ( 1 ) and rearranging, it is 
found that 


s = kSRc/i(Rc - SRj) 1—360 (13) 

Finally, on substituting for i its value given by 
equation 1 — 359(4), we are able to express the ac- 
tivity sensitivity entirely in terms of the known 
circuit parameters. Thus, 

^ kSg„.NpN,RiRa 
® (1 -h k) (R. - SR.) 

or 1—360 (14) 

s = kSg^NpN^R./d + k)[m- S(m - 1)] 

The reciprocal of s can be considered the percent- 
age gain in the output voltage (or in Ep or E.) for 
a unit percentage change in the gain of the tube, 
since dGa/G, is equal to — di/i. In the equation for 
s, note that if the thermistor sensitivity were 
equal to R,./Ri, the stability mathematically would 
be infinite. Since Rc/R. is greater than 1 , a single 
tungsten lamp could not provide the thermistor 
sensitivity for the above condition to hold unless 
special measures were available to reduce the heat 
leakage from the filament. The effective sensitivity 
could be increased if R, were replaced by another 
tungsten lamp, and the crystal unit were inserted 
in the place of R,. Theoretically, the sensitivity can 
be made much larger than unity simply by vary- 
ing the ambient temperature together with the op- 
erating temperature of the filament ; for instance, 
by constructing a thermistor with the filament 
mounted inside a heater sleeve and controlling the 
heater current by feedback from a later amplifier 
stage. If equation (14) is taken apart, it will be 
found that the denominator term, (Rc — SR.), 
originates from that component of dE, that is 
equal to — dE„. When there is an increase in E„ 
the voltage E, changes in two ways: one is due 
to the change in the current through R,, and the 
other is due to the increase in the resistance, itself. 
It is the latter component that is approximately 
equal to — dE,. Mathematically, the change in E, 
is expressed by the differential equation 


By equation (7) we see that dE,/dE. = 

E„(l — s)/E.. On substitution in (11) and after 1 350 (15) 


WADC TR 56-156 


231 



Section I 

Crystal Oscillators 

Since dR4/R4 is equal to SdE4/E4, on substitution 
in equation (15) it can be shown that 

dR4 = S 1-360 (16) 

If S is greater than 1, an increase in voltage across 
R4 must result in a decrease in current. (Inciden- 
tally, since the change in R4 is actually due to a 
change in the temperature brought about by an 
increase in power, a value of S greater than unity 
implies that the percentage increase in resistance 
is at least equal to twice the percentage decrease 
in current.) Now, assuming that the cujrrent 
through the input transformer is negligible, 
E, = I4RT, and dE, = l4dR4 -f RTdl4, where dRi = 
dRi. If E, is to remain constant, that is, if dE, is 
to equal zero for a small change in the gain of the 

dR 

tube, dh/U must equal — If the latter value 

Xv^ 

is substituted in equation (16), it will be found 
that for conditions of s = 00 : 


S = 


R*!* 

Rt — R4 


Rt _ Rc 

R3 Ri 


1—360 (17) 


This is the explanation of the term (Rc — SRi) 
in the denominator of equation (14). Other than 
the assumption that the changes in the current I» 
through the grid transformer can be considered 
negligible in their effect upon dE4, the term is 
enticely a function of the Rj and R4 arms of the 
bridge, and is not related to the gain characteris- 
tics of the rest of the circuit. No experimental 
data is available concerning the operation of the 
Meacham bridge oscillator with values of S greater 
than unity, when R, behaves as a negative resist- 
ance (an increase in Ej is accompanied by a de- 
crease in I4). Theoretically, if S were greater than 
Rc/Ri, an increase in the gn, of the tube would 
ultimately result in a decrease in the output- volt- 
age and in the voltage applied across the bridge. 
In an actual circuit, whether stable values of R4 
would be maintained under such conditions is open 
to question. Perhaps the thermal lag of the fila- 
ment and the extreme sensitivity of E* would so 
influence the operation that R, would periodically 
overshoot its mark and prevent an unmodulated 
equilibrium from being reached. In practice, the 
values of S will be on the order 0.5, so such con- 
siderations do not arise. For s to be as large as 
possible, referring to equation (14), it can be seen 
that [k/ (1 -f- k) ] should be as large as practicable. 
This agrees with the equations for frequency sta- 
bility if the circuit is to be designed according to 
equation 1 — 359(7). The term [k/(l -j- k)] has 


no maximum, but approaches unity as k increases 
indefinitely. Assume that s is equal to 50. This 
means that a change in the gain of the tube of 
1 per cent will cause a change of only one-fiftieth 
of 1 per cent in the output voltage. Or in terms of 
db, since 


s = 


E. 

dE. 


di 

i 


A (log i) 
A(log E.) 


A(log Gt) Adb in tube gain 

A(log Ep) Adb in output 

an increase of 0.5 db in the gain of the tube will 
cause only a 0.01-db increase in the output. 

CRYSTAL DRIVE LEVEL CONSIDERATIONS 
IN MEACHAM BRIDGE OSCILLATOR 

1-361. A starting consideration in the design of a 
Meacham bridge oscillator is that the crystal unit 
is not to be overdriven. If Pj is the crystal power, 

Ic = VPi/Ri is the crystal current, and 

mRi / Pi m-y/ Pi R, 


E, I- Rn 


^ UL - 

-i\ Ri 


m 


1—361 (1) 


With Ri determined by . the crystal unit, it is de- 
sirable, from the point of view of frequency sta- 
bility, for m, and hence Rz to have small optimum 
values. If S approaches unity, the small Rj will 
also be an important consideration in activity sta- 
bility, but for normal values of S the activity 
stability is improved slightly if Rj is large. The 


(ro - SR,) ' 


in equation 1 — 360 (14) has no 


term . 

,Ro - SR,. 

maximum, but approaches unity as Rj is increased 
indefinitely and R, and S are held constant. Usu- 
ally, the requirements of frequency stability are 
the more important, and Rz should be kept as small 
as practical thermistor resistances and values of 
k permit. At low frequencies, values of R, may be 
in the neighborhood of 1000 ohms or more. The 
voltage across the thermistor will be 


T?, E, _ le Rj _ E] _ ■%/ Pi Ri 

n;4 — ; ^ ; — 

m m — 1 m — 1 m — 1 

1—361 (2) 

where Ei is the voltage across the crystal unit. For 
convenience, we repeat equation 1 — 360 (1), but ex- 
pressed as a function of m and k: 


WADC TR 56-T56 


233 


Ep = (m — 1) i E,/km“ 


1—361 (3) 



section I 
Crystql Oscillators 


The power dissipation in R 4 is 


P« = £414 = E, I./(m - 1) k = P,/(m - 1) k 


(max) E 4 = 


~\/ Pcm Pm 
(m — l)\/ir 


1—362 (1) 


1—361 (4) 

The impedance of the bridge in terms of R, is 

Z. = R. Rt/(Rc + Rt) = kmRi/(l + k) (m - 1 ) 

1—361 (5) 

The plate impedance of the tube is 


Zp = Np» Z. = 


The plate voltage is 


Np* kmR, 


(H-k)(m-l) 


1—361 ( 6 ) 


E. 


= Ip Zp = 8 m Np^kmRi 

” " (1 + k) (m - 1 ) 


(7) 


Also, 


Ep = Np E. = mNpVPT^ /(m - 1) 

1—361 ( 8 ) 


and 


E. = N, Eo 1—361 (9) 

Finally, we repeat equation 1 — 359 (3), the over-all 
equation for feed-back equilibrium, but expressed in 
tmns of Ri, m, and k: 

G, G2 G, G4 = 8m Np i Ri ^ 

m (1 •+■ k) 

Rj can be adjusted to provide the same value of m 
for each different crystal unit. Under these cir- 
cumstances, E, and Ep will be the same in each 
oscillator, even though Ri varies. Two fundamental 
problems are that the design must ensure that the 
crystal current does not overdrive the crystal unit 
when R, is small, and that the thermistor current 
is sufficient for S to be a maximum. 

DESIGN PROCEDURE FOR MEACHAM 
BRIDGE OSCILLATOR 

1-S62. The fixed point of reference for estimating 
the current and voltage at any point in the 
Meaeham circuit is the thermistor voltage E^. 
This is the voltage that is required to make 
R* = R,/(m — 1). If R, and m are held constant, 
Et as well as E, (= mEt) and E, ( = NpE,) will 
also be constant. If Pnn is the rated crystsd power, 
Rm is the maximum series resistance of the crystal 

I> 

unit, and is the minimum expected resistance 

of the crys^ unit, then, by equation 1 — 361 (2), 
E 4 must not be greater than the value 


Since the Meaeham oscillator is most applicable 
for use in the low-frequency range where crystal 
units having resistances in the neighborhood one 
or more thousand ohms are not uncommon, the 
risk is greatly increased that an exceptionally 
well-mounted crystal will have a resistance of as 
little, as, perhaps, Rm/25. Also, since the Meaeham 
circuit is primarily useful as a precision oscilla- 
tor, an additional safety factor should be allowed 
to prevent the crystal unit from being driven be- 
yond its test specifications. For these reasons, it 
is suggested that in the absence of prior experi- 
ence or manufacturer’s recommendations for a 
given type of crystal unit, the Meaeham design for 
frequencies below 200 kc assume a minimum R 
of Rm/25, rather than Rm/9 as was assumed in the 
case of the parallel-resonant oscillator design. 
However, it can still be assumed that the most 
probable crystal unit will have an R = Rm/8. If 
the crystal unit to be used is a precision GT cut, 
a safety factor as large as N = 25 neW not be 
made. In any event, crystal units having resist- 
ances less than Rm/9 can be expected to be ex- 
tremely rare, and if N = 9 is considered a suffi- 
cient safety factor, an output voltage two-thirds 
greater can be realized than if N is assumed to 
be 25. A crystal unit having a resistance less than 
Rin/9 would be driven beyond its test level, but 
far below a level that could damage the crystal. 
Since the resistance is already low, an increase 
in resistance with overdrive would do more good 
than harm. The only concern is that the frequency 
of a borderline crystal may deviate beyond the 
tolerance limits. Such a risk could be checked dur- 
ing a production test, but would subtract from the 
reliability of crystal replacements in the field. In 
equation 1 — 361 (10), it can be seen that when k 
is a minimum (when R, = R„), the imbalance, 
as measured by i, is a minimum. When k is large, 
the percentage changes in (k 4 - 1 ) and in Ri are 
very nearly equal, so that the imbalance tends to 
vary as the square of R,. k should be chosen for 
maximum frequency stability under variations of 
C„ assuming that the crystal unit resistance is its 
most probable value (approximately Rm/3). Ac- 
cording to equation 1 — 359 (5), with all else fixed, 
the percentage change in frequency is a minimum 
when k = 1. The most probable optimum value of 
k, therefore, fixes R, as equal to Rm/3. Thus, for 
any random value of R„ 

k = Rm/3R, 1—862 (2) 


WADC TR M-156 


233 



Section I 

Cryttal Oscillators 


Next, a value of m equal to 4/3 (see equation 
1 — 359 (10) ) should be chosen, to provide maxi- 
mum frequency stability on the assumption that 
equations 1 — 359 (8) and (9) are to apply when 
R, is its most probable value. After this is done, 
a safety factor of N should be selected, and the 
maximum value of E4 should be determined by 
equation (1) , such that it will not require a bridge 
voltage sufficient to overdrive the crystal unit 
when R, is equal to R.„/N. Next, a thermistor is 

R 

chosen that will provide a value of R, = ^ 

m — 1 

when Ej is equal to, or less than, the maximum 
, value determined above. Next, the ratio Np/N, can 
be determined, using equation 1—359 (7) with the 
assumption that k = 1 and m = 4/3. This gives 
N 

a ratio ^ = \/T5 «= 4, which value thus provides 

the greatest probability that random changes in 
Cp and C,, can cancel when R, is its most probable 
value. The next step is to select a tube with hig^i 


class-A gm and R„. A 6AC7 would be very satisfac- 
tory. Using equation 1 — 361 (6), determine Np on 
R R 

the assumption that Z„ = when R, = Now, 

Np can be made equal to N|,/4. R;, of course, must 
be variable over a percentage range comparable to 
that to be expected from the crystal unit. Normal 
tube voltages are used. The other circuit compo- 
nents can be determined according to the tube 
specifications for class-A operation and the special 
output requirements of the oscillator. Ep, E„ E„, 
Ip, I,., etc can be determined from the equations in 
paragraph 1-361, the frequency stability from 
equations 1 — 359 (5) and (6), and the activity 
stability from equation 1 — 360 (14) for maxi- 
mum, most probable, and minimum values of R,. 

MODIFICATIONS OF MEACHAM BRIDGE 
DESIGN 

1-363. Two designs of the Meacham bridge sta- 
bilized oscillator are shown in figure 1-163. In each 



Fig. 

Equipment 

Purpoae 

F, 

CR 

(A) 

Control-Monitor 

IP-68/CPN-2A 

Timing osc. Con- 
trola indicator 
sweep freq and prr 
of shoran station 

186.22 

Oven 

controlled 

(B) 

Radio Set 
AN/FRC-10 

Carrier osc and 
phase-shift cir- 
cuit 

100 

WECo 

D-163897 

D-169649 


Ri 

Rs 

Rs 

Rt 

0.1 

Thermi- 

stor 

250 

0.12 , 



Circuit Data for Figure 1-163. F in kc. R in kilohms. C in iiiti. h in jih. 


234 


WADC TR 56-1S6 




























! ■< * 


Section I 
Oyticil Oscilloton 


of these circuits inductor-capacitor combination 
has been connected in series with the crystal unit. 
Obviously the combination is intended to be reso- 
nant at the crystal frequency. The variable ar- 
rangement shown in figure 1-163 (A) permits the 
frequency to be pulled to a more exact value if 
desired, the crystal unit (if necessary) operating 
with a reactive component in its impedance. Or, in 
case the tube operates into a partly reactive load, 
the tuning elements in the bridge could permit the 
crystal, itself, to operate at exactly series reso- 
nance. The series inductor and capacitor are effec- 
tive in aiding the initial build up of oscillations 
and in ensuring that the crystal assumes control 
at the frequency of the desired mode. It can also 
be presumed that the LC combination in the bridge 
improves the waveform somewhat and reduces the 
small distortion introduced by the tungsten lamp. 
This distortion is due to the fact that the filament 
cools at least to some extent during the time that 


the current alternates from its effective value in 
one direction to its effective value in the opposite 
direction. At frequencies above 100 cycles per sec- 
ond this distortion in the waveform is not serious. 
At radio frequencies it is normally small compared 
with the distortion introduced by the tube. The 
resistance R; in figure 1-163 (A) appears to be 
inserted in order to maintain a constant tube load 
by minimizing the variations in the bridge im- 
pedance due to adjustments and to crystal units 
having difference resistances. In figure 1-163 (B) 
note that the crystal unit is grounded. This is the 
usual arrangement. The parallel primary wind- 
ings of the grid transformer in the same figure 
suggest that the arrangement is an adaption of a 
readily available transformer, very probably of 
the same construction as the one in the plate cir- 
cuit. The parallel primary connection is in the di- 
rection of phase addition. Because the near-unity 
coupling between the coils effectively doubles the 




R. 

Rt 

c, 

c . 

c, 


c. 

c» 

c ^ 

VI 


22 

■ 

0.56 

1 

no 

40 

300 

100,000 

10,000 

300 

10,000 

6AU6 

2651 

0.15 

0.072 

■ 

140 

140 

100 

100 

500,000 

500,000 

500,000 




WADC TR 56-156 


235 





























Section I 

Crystal Oscillator* 

inductance of each, the parallel connection pro- 
vides the same step-up arrangement and primary 
impedance that would be provided by only one of 
the coils if used alone, but with a reduction in the 
leakage inductance. A Meacham bridge-stabilized 
oscillator can be designed employing two or more 
tubes. On the average, slightly better frequency 
stability can be achieved with a two-tube circuit, 
but only in rare instances are the additional cost, 
space, and weight requirements worth the small 
improvement in performance. Perhaps, at fre- 
quencies in the neighborhood 1000 kc or higher the 
two-tube arrangement could be more profitable 
than the one-tube stage. The design of a multi- 
stage bridge oscillator can be practically the same 
as that of the one-stage circuit except that the 
tube gain, Gj, is replaced by G 21 G 2 J . . . Gs„, where 
Gzk is the voltage gain of a transformation stage 
between the output and input of the bridge, and 
where n is the total number of such transforma- 
tions. By increasing the number of positive-db 
stages, the bridge i can be made as small as de- 
sired, and the frequency and activity stability will 
be increased in proportion to the gain. It is because 
the possible gain is unlimited for all practical pur- 
poses that the Meacham oscillator represents the 
ultimate in precision control of the frequency. In 
the final analysis the limiting condition is the de- 
gree to which the crystal parameters, themselves, 
can be kept constant. Figure 1-164 shows the basic 
circuit of a two-tube Meacham oscillator that em- 
ploys no transformers and offers the advantage 
of only a single tuned stage. Although the design 
equations are somewhat different from those of 
the conventional one-tube stage, the same basic 
approach is to be employed, and the problems to 
be encountered can be solved similarly to those of 
the transformer-coupled circuits. 


Capaeitance-lrid9e Oscillators 

1-364. Capacitance-bridge oscillators may po.ssibly 
prove suitable for use in the v-h-f range. Their 
advantage lies in the fact that a properly balanced 
capacitance bridge cannot provide sufficient feed- 
back of the proper phase to sustain oscillations at 
any frequency other than the tuned frequency of 
the circuit, provided a crystal unit is connected 
in the circuit that has a resonant mode of vibra^ 
tion at the tuned frequency. A properly balanc^ 
capacitance-bridge oscillator is thus crystal-con- 
trolled, rather than crystal-stabilized. On the other • 
hand, if the bridge is not balanced, the circuit can 
operate as a free-running oscillator, which may 
or may not be crystal-stabilized. For the purpose 
of ensuring operation of crystal units at desig- 
nated very high harmonic modes, the capacitance 
bridge, if not the most dependable, is at least as 
dependable as any other so far tested. The princi- 
pal disadvantage of this type of circuit is that 
rather critical tuning adjustments must be made, '^« 
and one crystal unit cannot replace another unless 
these adjustments are repeated. Largely on this 
account the circuit is not to be preferred for fre- 
quencies below 50 me, and perhaps not below 76/y 
me. Nevertheless, once the bridge is properly ad-^ 
justed, the operation with a crystal unit free of 
spurious modes is dependable under any extremes 
in temperature that can be reasonably expected. 

BASIC CIRCUIT OF CAPACITANCE- 
BRIDGE OSCILLATOR 

*1-365. Figure 1-165 illustrates the basic circuit 

* The discussion in paragraph 1-36S is based upon the 
analysis of the basic circuit app^ring in the report, 
H.F. Harmonic Crystal Investigation, by S. A. Robinson 
and F. N. Barry of Philco Corporation, on Army Contract 
#W3S-038 8C-14172, 1947. 



WADC TR 56-156 


figvn 1-164, Two-slago Moaeftom brUgm-stablllxnd osciMotor 

336 



B-f 


Figure 1-165. (A) Basic circuit of capacitance-bridge 
oscillatar. (Bi Equivalent circuit of capacitance- 
bridge oscillator 

of a capacitance-bridge oscillator. The first design 
consideration in this oscillator, as in all others 
that are to operate in the v-h-f range, is to keep 
all the leads as short as possible. Those circuit ar- 
rangements and circuit components that provide 
a minimum of stray inductance and capacitance 
are to be preferred. In the circuit of figure 1-165 
(A), Lp and L, are actually a single, tapped induc- 
tor with the two sides wound on the same form 
and tightly coupled together. The induced-voltage 
effect is equivalent to that of a single generator 
connected across both coils and driving the bridge 
with an emf (Ep -f E,) Ep(Np -1- 1). Np, the 
turns ratio of Lp to L„ is usually, and most con- 
veniently, equal to 1. In case the shunt capacitance 
of the crystal unit is greater than 10 it would 
be desirable to make Np slightly greater than 1. 
An Np greater than 1 but less than 2 can be ex- 
pected to provide a higher output, but the opera- 
tion will tend to be more critical and the frequency 
less stable. Before the circuit is placed in opera- 
tion, the bridge must be balanced at an off-reso- 
nance frequency, so that no voltage can appear 
across the grid circuit. At an off-resonance fre- 
quency the crystal unit appears as a capacitance 
Co, so that under the conditions of balance 



1-^65 ( 1 ) 

With Np ^ 1, Cb is adjusted to equal C*. (Cp is 
here assumed to include Cpp, and Lp to account for 
Cp. See figure 1-165 (B).) The total capacitance 


Section I 
Crystal Oscillators 



(B) 

in the circuit is thus. 


Ct = Cl + 1—365 (2) 

2 

Since the distributed inductance of the crystal 
leads, L', tends to increase the effective value of 
Co, the frequency at which the bridge is balanced 
should not be greatly different from the intended 
operating frequency. If L' is unduly large, a series 
capacitance should be connected in the crystal arm 
of the bridge sufficient to annul the stray induct- 
ance in the vicinity of the operating frequency. 
Once the bridge is balanced, Cp should not be ad- 
justed again. The initial adjustment is rather 
critical, requiring an accuracy of a few tenths of 
a micromicrofarad. Co and Cp in the v-h-f range 
should be as small as possible. The coaxially- 
mounted crystals, such as those contained in the 
HC-IO/U holder, are to be preferred on this ac- 
count. Values of C„ in the neighborhood of 4 or 5 
/ifif are quite feasible. Co can be further reduced 
by connecting an inductor across the crystal unit 
to annul part, but not all, of the shunt capacitance ; 
however, this should be avoided, because the pres- 
ence of the inductor would narrow the frequency 



WADC TR 56-156 


237 



Section I 

Crystal OKillators 

range over which the bridge can be considered 
balanced. Cb must be adjustable over the expected 
capacitance range of the particular type of crys- 
tal unit to be used. C, is the tuning capacitance. 
For crystal-controlled operation, Ci is adjusted so 
that the total circuit capacitance Ct is approxi- 
mately resonant with the total inductance at the 
operating frequency. To balance the circuit, C, is 
set to a position that tunes the bridge to a fre- 
quency well off the resonance frequency of the 
crystal unit. Referring to figure 1-165 (C), it can 
be seen that E, equals (E, + Ecb)- At the tuned 
frequency, R can be neglected and the crystal unit 
considered as a capacitance, C,. Approximately, 
E, and Ecr are 180 degrees out of phase, and there- 
fore tend to annul each other. Now, assiune that 
Cb is made to approach zero. 1 , and Ecr therefore 
become negligible, and the circuit behaves as if 
the crystal side of the bridge were open-circuited 
at Cb. The remaining circuit would be simply a 
Hartley oscillator with the crystal unit serving as 
a blocking capacitor between the inductor and the 
grid. If Cb is gradually increased, Ecr builds up 
until a point is reached where E, effectively is can- 
celed and E( is insufficient to sustain oscillations. 
Cb should then be increased one more increment 
beyond the oscillation cutoff. At this setting of Cb, 
the bridge can be considered properly balanced, 
but a check should first be made that oscillations 
do not occur at other settings of C, well removed 
from its value for crystal control. If such oscilla- 
tions do occur, the adjustment of Cb should be 
repeated. The free-running oscillations can be dis- 
tinguished from the crystal-controlled oscillations 
by the continuous nature of their activity curves 
as measured by grid current and output meters 
when Ct is varied above and below a discontinuous 
region. A discontinuous point indicates an abrupt 


Ip-r|+l2+l3+l4 



figun 1-165. (C> SimpMlod •quivafent circuit 


change to crystal control, where the frequency be- 
gins to change at a much slower rate with varia- 
tions in circuit capacitance. However, once the 
bridge is balanced, no oscillations occur except 
near the crystal resonance frequency, in which 
region the bridge balance is upset. 

1-366. Referring again to figure 1-165 (C), with 
the circuit balanced, suppose that C, is gradually 
increased from its minimum value. At some point 
oscillations suddenly start; as Ci is further in- 
creased, the activity builds up to a maximum and 
then sharply declines, as is illustrated in figure 
1-166. Note also the sharp decrease in frequency 
when maximum amplitude is approached. Appar- 
ently, when oscillations first begin, the crystal ap- 
pears inductive. Ecr therefore has a large compo- 
nent in phase with E„ and the circuit is essentially 
a modified Miller oscillator. Also, the ratio of I 4 to 
I, is a maximum, since Ecr tends to cancel the 
voltage across Cb. As C, is slowly increased, the 
frequency and the inductive reactance of the crys- 
tal drop. This means that the effective Q of the 
grid circuit also decreases. Although the presence 
of the capacitance C, modifies the phase relations, 
the circuit performs fundamentally as a Miller 
oscillator. L, can be interpreted as something of 
a booster inductor to increase the effective induct- 
ance of the crystal unit, and 1 ^ can similarly be 
viewed as a booster current to boost the voltage 
across the inductive component in the grid circuit 
without, at the same time, increasing the voltage 
across the crystal R*. That the capacitance-bridge 
circuit actually has the same characteristics as 
does the Miller circuit is well illustrated by the 
similarities between the curves of figure 1-166 and 
the equivalent curves for the Miller oscillator 
shown in figure 1-144. Note that for both oscilla- 
tors, the circuit capacitance for maximum excita- 
tion does not coincide with, but is smaller than, 
the value for maximum output. One significant 
difference between the two circuits is the fact that 
the Miller circuit cannot maintain the proper feed- 
back phase if the crystal is operated at series 
resonance, whereas the capacitance-bridge circuit 
can, because of the presence of L,. Where oscilla- 
tion cutoff for the Miller circuit is above the series- 
resonance frequency of the crystal, it is below the 
series-resonance frequency in the capacitance- 
bridge circuit. 

1-367. If the crystal control is to be fully effective, 
the series-arm resistance must be small compared 
with the shunt reactance, Xco- A’thougi. this 
requirement becomes increa’ingl' diffi lit at 
the higher harmonics, it te avhievect, uven 
at frequencies well above lOc m' . Assuming that 


WADC TR 56-156 


238 



Section I 
Crytlal Otcillalon 




FIgun 1-166. Typical performance curves of capacitance-bridge oscillator, showing effects of change In bridge 
tuning capacitance on voltage output, activity (d-c grid current), and frequency 


the series-arm R is not more than one-tenth 
the magnitude of Xc, then the approximate 
equation for the effective crystal reactance, X, = 
X.Xco/(Xco -|- X,), where X, = 4irLAf series-arm 
reactance, is sufficiently close for an interpretation 
of the capacitance-bridge performance. Now, os- 
cillations cannot start unless |Xcb| > X* -t- Xl,. 
Xcb we shall assume is equal to Xco under the con- 
ditions of balance. X, is equal to |Xco|, and hence 
to |Xcb|, when X, = — Xob/2, that is, when the crys- 
tal unit is halfway between series resonance and 
antiresonance. Thus, when oscillations start, the 
crystal frequency must be much nearer to the reso- 
nant than to the antiresonant state. Also, the plate 
circuit must appear inductive to the vacuum tube 
to a degree dependent upon the effective Q of the 
grid circuit. This means that Ii must be slightly 
greater than (I* -f- Is + I 4 ). In figure 1-165 (A), 
it can be seen that the crystal unit operates into 
a load reactance approximately equal to the paral- 
lel combination of C, and the inductor (Lp -f L.) 
in series with Cb. As the reactance of Cj approaches 
that of the inductor, the reactance of the parallel 
combination rises very sharply, and a small change 
in C, can make a large change in the load reactance 
across the crystal unit. More than any other factor, 
this is the reason for the sharp dip in the fre- 
quency curve as C, approaches a maximum. 
1-368. It is not possible to tell at which point in 


the curve the crystal passes through series reso- 
nance. Since at series resonance the reactance of 
C, in parallel with L„ and L, is equal, approxi- 
mately, to — Xcb, the resonance frequency may well 
be below the knee of the curve for a crystal having 
a very small Co (conditions for large Xcb and near- 
parallel resonance of C, with the inductor) and 
above the knee for crystals of larger At 
series resonance, if R is small h approximately 
equals laCb/Ci. Assuming that E.(= I.Xl,) leads 
Ecb(= RR) by 90 degrees, the effective Q of the 
grid circuit at series resonance is equal to E,/Eck. 
When Eo and Ecr are expressed as functions of I 3 , 
Cb, C„ and Xi,„ it can be shown that (series reso- 
nance) Qp = ^ = Xi..(C,-^Cb) 

equation is only a broad approximation in the y-h-f 
range, since all the distributed parameters have 
been ignored, particularly the grid capacitance 
and the resistance of the inductor. However, it 
does indicate that the larger the ratio of C, to Cb, 
or, equivalently, Xco/Xci, the smaller the inductive 
phase shift that will be required in E,„ and the 
more nearly will the bridge tank approach parallel 
resonance. If R, or rather the total grid losses, 
should increase or decrease, the frequency will de- 
crease or increase, respectively. It seems safe to 
assume that crystal units having the larger values 
of RCo products will operate fairly near their 


WADC TR 56-156 


239 



Section I 

Cryctal Otcillolor* 

series-resonant state. This is due partly to the fact 
that the smaller the Xoo/R ratio, the smaller the 
frequency range between resonance and antires- 
onance. Crystal units having the smaller values 
of RCo will perform with greater amplitude and 
frequency stability if operated above series reso- 
nance. Unfortunately, crystal units in the v-h-f 
range are tested only for series resonance. The 
greater likelihood of the occurrence of unwanted 
modes increases the importance of having the cir- 
cuit designed so that the operation of the crystal 
unit lies within its tested specifications. While the 
capacitance bridge is excellent for preventing all 
modes of oscillation except the one desired, it is 
not a true series-mode oscillator, although it is 
so classified because its v-h-f application requires 
the use of crystals that are only series-tested. 
Rather, the oscillator is something of a hybrid 
between a Miller and a stabilized Hartley circuit. 
In the interest of frequency and amplitude sta- 
bility, the circuit should be adjusted to operate 
above the knee of the frequency curve. A setting 
of the tuning capacitance corresponding to a grid 
current of 50 per cent of the maximum possible 
provides, approximately, the optimum output volt- 
age and operating state nearest series resonance 
that are consistent with the operating region of 
better stability. The peak of the voltage-output 
curve in figure 1-166 corresponds closely to the 
adjustment for maximum tank impedance, which 
certainly occurs below series resonance where the 
crystal unit appears as a capacitance. The larger 
the capacitive reactance that the crystal unit can 
have and still permit oscillations, the more nearly 
can series-resonance oscillations fall within the 
higher stability region. For this purpose, the ratio 
of L„ to Lp and of Ch to C„ should be as large as 
unity, or greater, when the capacitance-bridge 
oscillator is to be used with series-tested crystal 
units. 

DESIGN MODELS FOR CAPACITANCE- 
BRIDGE OSCILLATORS 

1-369. The circuits shown in figures 1-167 through 
1-171 represent five different modifications of the 
capacitance-bridge oscillator. These circuits were 
designed and tested by the research team of S. A. 
Robinson and F. N. Barry of Philco Corporation. 
No single type of circuit was found to be superior 
for operation over the entire tested frequency 
range of 50 to 200 me, but each circuit has advan- 
tages for certain applications. The inductive arms 
of the bridge can be a single, self-supporting 
tapped inductor having an inside diameter of one- 
quarter inch or greater. Silver-plated AWG No. 16 


wire can be used. The tuning and balancing capaci- 
tances are small air capacitors. The fixed capaci- 
tances are, for the most part, the button-mica Erie 
type. Composition resistances are used, having 
nominal values of ±10 per cent. Successful opera- 
tion of any of the circuits depends largely upon 
arranging the circuit components to permit the 
shortest possible leads, and all components should 
be of small physical size. Silver-plating of the com- 
ponents is desirable, and careful shielding and the 
use of low-loss insulating materials is necessary. 
Without good shielding and well-insulated capaci- 
tor shafts, it may be impossible to adjust the 
bridge properly because of the effects of hand ca- 
pacitance. Transmit-time effects become quite sig- 
nificant as the frequency is increased beyond 50 
me. The lag in the response of the plate current 
with rapid changes in the grid voltages is equiva- 
lent to the circuit behavior that would result if 
an inductance were connected in series with R,, of 
the tube. The lower the plate voltage, the larger 
is the apparent inductance and its accompanying 
tendency to lower the frequency. Usually, this 
effect makes it easier to operate the crystal unit 
at series resonance, but the need for careful B + 
regulation becomes all the more important. For 
normal voltages, transit lag is approximately 0.2 
degree per megacycle in v-h-f tubes such as the 
6AK5. 

SINGLE-TUBE, 50- TO 90-MC 
CAPACITANCE-BRIDGE OSCILLATOR 

1-370. The circuit shown in figure 1-167 has been 
operated at frequencies as high as 135 me, but its 
particular merit lies in its performance at frequen- 
cies between 50 and 90 me. When operated in the 
high-stability region, output up to 10 volts can be 
obtained, although care should be taken that the 
rated drive level of the crystal is not exceeded. 
Outputs of 2 milliwatts into an inductively cou- 
pled 100-ohm resistor can be obtained in the same 
operating region. 

COMPACT, MINIATURE, 50- TO 120-MC 
CAPACITANCE-BRIDGE OSCILLATOR 
1-371. The circuit shown in figure 1-168 is par- 
ticularly suited for construction as a small, pack- 
aged, plug-in oscillator. If desired, several such 
oscillators of different frequencies can be designed 
as interchangeable units of the associated equip- 
ment. The entire shielded unit need not occupy a 
space greater than 6 cubic inches. The maximum 
frequency at which this circuit was found to oscil- 
late was 156 me, but the activity at that frequency 
was less than one-tenth that at 50 me. At 120 me 
the activity is approximately one-fourth of that at 


WADC TR 56-156 


240 


Section I 
Crystal Oscillators 



Figure 1-167. A eingle-tube capacitance-bridge oscillator which is practical for operation in the 50—90-mc 
frequency range. Resistors not otherwise specified are Vi w. L-1 and 1-2 
are a single center-tapped ceil of suitable inductance 



Figure 1-168. A plug-in capacitance-bridge oscillater which is practical for 50-150-mc frequency range 


WADC TR 56-156 


241 



Section I 

Crystal Oscillators 


50 me, so 120 me appears to be the most praetieal 
upper frequeney limit. A subminiature tube hav- 
ing high transeonduetance is used. Greater output 
is to be aehieved with a triode, but greater fre- 
quency stability is to be had with a pentode. With 
a triode, the comparatively large plate-to-grid ca- 
pacitance which shunts the balancing condenser 
may make it difficult to achieve a balancing capaci- 
tance as small as that of the crystal unit. This 
condition requires that the L-2A section of the 
bridge inductor be somewhat larger than the L-2B 
section. The possible output is reduced thereby, 
but the crystal unit will be operated nearer its 
series-resonance frequency. The output secondary, 
L,, can be a single turn coupled to the plate end 
of L,. 

CAPACITANCE-BRIDGE OSCILLATOR FOR 
GREATER POWER OUTPUT IN THE 
50- TO 80-MC RANGE 

1-372. The circuit shown in figure 1-169 was de- 
signed for the purpose of achieving a maximum 
power output without regard to the rated drive 
level of the crystal unit. Ho'wever, none of the 
crystals used were fractured during the experi- 
ments. The higher-power circuit is essentially the 
same as that of figure 1-167 except that the N,, 
ratio of the bridge inductor is greater, higher volt- 
ages are used, and a 6AG7 replaces the 6AG5 tube. 
Although the 6AG7 has a higher transconductance 
and power rating than the 6AG5, the interelec- 
trode capacitances are greater, the internal leads 
are longer, and the base is constructed of higher- 
loss material. The circuit operated at frequencies 
as high as 102 me, but above 80 me the disadvan- 
tages introduced by the vacuum-tube construction 
make the circuit impractical. Better performance 
might be expected with a 6AH6. With the tube 
operated near its maximum rated dissipation, a 
one-watt inductively coupled output was obtained 
at 54 me, and one-third watt at 80 me. These out- 
puts are representative of the peak obtainable. 
Much less power is to be had if the oscillator is ad- 
justed for operation in the higher-stability region. 

TWO-TUBE, 50- TO 100-MC 
CAPACITANCE-BRIDGE OSCILLATOR 

1-373. The circuit shown in figure 1-170 is similar 
in operation to the one-tube circuit except that the 
feedback has an additional antplifier stage to boost 
the gain. There is a significant difference in that 
the crystal unit is connected to the plate aide of the 
bridge. Under this arrangement, the excitation 
voltage of V, lags the r-f plate voltage of V„ 
which means that if the plate load is resistive the 


r-f plate voltage of V, would tend to lag the re- 
quired excitation voltage of V.,. For oscillations 
to occur, the plate tuning tank of V, must appear 
inductive in order to shift the input of V. to the 
proper phase. After equilibrium is reached, a 
slight increase in the value of C, causes the plate 
impedance of V, to become more nearly resistive, 
and therefore the input of V. becomes more 
nearly 180 degrees out of phase with the input of 
V,. This requires that the frequency drop to a 
point where the voltage across C ^ is more nearly 
in phase with the r-f plate voltage for V,. For 
both tubes to operate into resistive loads, the 
crystal unit must appear as a capacitance. For the 
crystal unit to operate near series resonance and 
at the same time maintain the oscillations in the 
higher-stability region, it would seem that R,, the 
parasitic damping resistor in the input circuit of 
V. can be replaced, if necessary, by a resistance 
comparable in value to the V. input reactance. 
The effect will be to shift the input phase in 
a lagging direction, which would require the V, 
tank to be more detuned, and hence less critically 
adjusted. This, in turn, will require a comparable 
shift in the phase of the input to V„ which is to 
be had by a decrease in frequency, thereby per- 
mitting the bridge to be less critically tuned in the 
vicinity of the crystal resonance point. The circuit 
in figure 1-170 was found quite practical for use 
as a test oscillator for measuring the relative per- 
formance characteristics of harmonic-mode crys- 
tal units. During the temperature runs, even 
though frost had collected on various components, 
the operation of the circuit was little affected. For 
duplicate units of this circuit to provide essentially 
the same meter readings for tests of the same 
crystal unit, it is necessary that the vacuum tubes 
used in the twin circuits show the same plate char- 
acteristics within ±5 per cent. A breadboard 
model of the oscillator having different values of 
tuning inductances was able to operate at 140 me. 
L„ in figure 1-170, is a 5-turn coil, approximately 
one-quarter inch in diameter; La and L., are the 
two halves of a 4-turn, center-tapped coil, approxi- 
mately one-half inch in diameter. 

MULTITUBE CAPACITANCE-BRIDGE 
OSCILLATOR OPERABLE AT FREQUENCIES 
UP TO AND ABOVE 200 MC 

1-374. The circuit shown in figure 1-171 has been 
used to generate crystal-controlled frequencies as 
high as 219 me, the seventy-third harmonic of a 
3-mc crystal. This frequency approaches the ulti- 
mate directly obtainable with quartz crystals at 
the present state of the art. A large part of the 


WADC TR 56-156 


242 



Section I 
Cryetal OKillalon 



Figure 1-169. A eapaeitanee-bridge oscillator tor higher power output which is practical for operation in the 
SOSO-mc frequency range. Resistors not otherwise specified are Vi w. All fixed capacitors have mica dieiectrics 



figure 1-170. A iwo-tube capacitance-bridge oscillator which Is practical 
for operation in the 50-l00>mc frequency range 


WADC TR 56-156 


243 





S«ction I 

Crytlol Osdilcitor* 


/ 



Figure 1-171. A multitube capacitance-bridge aeclllator which la practical for operation in the 50-200-mc range. 

Keaiatora not otherwise specified are rated at 'Aw 


success of the oscillator in figure 1-171 is due to 
the balanced electrical and mechanical nature of 
the push-pull capacitance-bridge circuit. The op- 
eration is very much the same as that of the cir- 
cuit in figure 1-170 except that the bridge stage is 
operated in push-pull. With different values of 
inductance, the circuit provides reliable frequency 
control anywhere in the v-h-f range, from 200 me 
on down. Probably its most practical application 
is as a harmonic test oscillator. The upper fre- 
quency obtainable is not limited by the circuit 
itself, but by the resistances and shunt capaci- 
tances of the crystal units. 

OTHER MODIFICATIONS OF THE 
CAPACITANCE-BRIDGE OSCILLATOR 
1-375. A number of capacitance-bridge modifica- 
tions have been successfully attempted, four of 
which are illustrated in figure 1-172. The circuits 
are largely self-explanatory, and will not be dis- 
cussed here. Probably of most importance is the 
electron-coupled circuit, since it permits frequency 
multiplication in the plate circuit. The triode con- 
nection of the crystal circuit probably prevents 
the crystal, itself, from being operated at frequen- 
cies above 75 me. 


The Butler Oteilioter 

1-376. At the present time, probably the most 
widely used of the series-mode oscillators is the 
Butler, cathode-coupled, two-stage oscillator. The 
basic design and equivalent circuits are shown in 
figure 1-173. Although the single-tube, trans- 
former-coupled type of oscillator will probably out- 
rank the two-tube circuit eventually, the Butler 
oscillator is the more popular at present because 
of its simplicity, versatility, frequency stability, 
and, of most importance, its comparative reliabil- 
ity. With the older types of crystal units, it was 
generally found that the Butler circuit was the 
least critical to design and to adjust for operation 
of the crystal at a given harmonic. The balanced 
arrangement of the circuit and the fact that twin 
triodes can be obtained in a single envelope con- 
tribute a saving in space and cost, and permit the 
use of short leads. For greater frequency stability 
than is normally to be had from parallel-mode 
oscillations, the cathode-coupled circuit can be 
used quite satisfactorily at any of the lower fre- 
quencies provided the resistance of the crystal 
unit is not greater than a few hundred ohms. 
However, the power output is small by comparison 


WADC TR 56-156 


244 




Figure 1-172. Miscellaneous 

with that of the Mille'r circuit for the same crystal 
power, and the broad bandwidth without plate 
tuning" of the Pierce circuit is not matched. The 
Butler circuit is usually designed for class-A op- 
eration, but class C is possible if greater output 
and plate efficiency are desired. The output may 
be taken from almost any part of the circuit — the 
plate or cathode of either tube. Quite often the 
cathode follower, V,, in figure 1-173, is a pentode, 
with the screen, control grid, and cathode forming 
a triode section electron-coupled to a plate circuit 
that usually is tuned for frequency multiplication, 
although the electron coupling can be employed 
simply to obtain greater output amplitude and to 
isolate the load from the oscillator circuit. At very 
high frequencies, where the shunt reactance of 
the crystal unit approaches the magnitude of the 
series-arm R, the operation is generally improved 
by shunting the crystal unit with an inductor that 


Section I 
CrycHii Oscillators 




capacitance-bridge oscillators 

is antiresonant with the shunt capacitance of the 
crystal at the operating frequency. When properly 
designed and adjusted, the two tubes operate 180 
degrees out of phase into resistive loads, and the 
crystal unit acts as a pure resistance. 

1-377. In figure 1-173, V, is connected as a 
grounded-plate cathode follower. The V, output 
current, I,., enters the feed-back path through the 
crystal unit, which is operating at series reso- 
nance. The impedance of the crystal unit is thus 
approximately equal to the equivalent series-arm 
resistance, Vj, a grounded-grid amplifier connected 
in the feed-back circuit, is excited by Ij, the com- 
ponent of I,, that passes through Rj. I,, 2 , the re- 
maining component of the feed-back current passes 
through Vj. It can be seen that the inpat voltage 
of V„ Eg,, is equal to the output voltage of V, if 
we assume that the coupling capacitance is infinite. 
The plate circuit of V, is broadly tuned to the de- 


WADC TR 56-156 


245 



Section I 

Crystal Oscillators 



II 


O. 

N 




WAOC TR 56.156 


246 


figurm 1-173. (A) Bosk diagram of Burtar two-stage cathode-coupled oseiilalor. (B) Equivalont r-f circuit of Butlor osciffator. Current arrows iadieato 
instantaneous electron Bow when r-f voltoges have poiarHtos shown. (C) SimpUBod equivalent circuit where gonorator of grounded-grid ampliBor Is 
replaced by a negative resistance. ID), (E), (f), and (G) Progressive simplHIcations el equivalent Butler tircult 



sired frequency. If Zp were simply a resistance, 
the circuit could still oscillate at the first crystal 
harmonic. If the crystal unit were shorted out, the 
circuit could also oscillate, but with the frequency 
controlled by the tuned plate circuit. R, and R, are 
usually equal, havintf values between. 50 and 200 
ohms. V, and V, are also usually of the same tube 
type. As will be seen, frequency stability is im- 
proved with largre values of transconductance. Note 
in figure 1-173 (B) that the r-f plate current in Vi 
is greater than that in Vj. As in all other vacuum- 
tube oscillators, there are two fundamental equi- 
librium conditions to consider: the over-all gain 
must equal unity, and the over-all phase shift must 
equal zero. We shall first consider the factors 
affecting loop-gain. 

LOOP GAIN OF BUTLER CIRCUIT 
1-378. At equilibrium we can say that 

1 Gj Ga (respectively) 

1—378 (1) 

The immediate problem is to find the values of Gk 
in terms of the circuit parameters. First, referring 
to figure 1-173 (ignore the capacitance C,ei in cir- 
cuit (B) ), assume that the voltage across R^ is 
approximately equal to Ep, then 

E,i = Ep - E, 1-378 (2) 


Section I 
Crystal OscHlotort 


Ip. = 


Ml t:i 

1 Hip 

Ml + 1 


R„ 


Mi + 1 


+ Zk 


Note that with Zk fixed by the external circuit, 
the plate current is related to the excitation volt- 
age, Ep, in such a way that the tube behaves as if 

it had an effective amplification factor of — 

Ml + 1 

R 

and an effective plate I'esistance equal to — 

Ml + 1 

This resistance is given the symbol Z, in figure 
1-173. If an additional resistance, Rl, were con- 
nected between the plate of Vi and r-f ground, Zi 

would equal ^ . Now, to find the value of 

Ml + 

G, = Eg,/E„ we start with 


E^ = I* Rj 1—378 (7) 

El = le R — Ma E*>+ Ip2 Rps + Ipa Zi, 

1—378 (8) 


I. = la + I, 


p2 


1—878 (9) 


and 


Ip» 


_ E ,2 Ma Ei 2 _ (M2 I) Ega 
Rpa *1" Zl Rpa "I" Zl 


1—378 (10) 


El = Ipi Zk 

and 


1—378 (8) 


Ip. 


M. E,i _ M. Ep — M.Ip.Zk 

Rpi + Zk Rpi + Zk 


M. Ep 

Rpi -b Zk (mi + 1) 


1—878 (4) 


On rearranging equation (10) to express E,, as 
a function of Ip„ and substituting this function for 
Efi in equation (8) , we have 

El = IpR + Ip*[- ^ + Zl] 

1—378 (11) 


On combining equations (8) and (4) 


El = 


Ml 


Ep Zk 


Rpi + Zk (mi + 1) 


and 


1—378 (5) 


G, 


Ei/E = ^ — gml_Zk — 

Rpi H- Zk (M. + 1) 1 + gml Zk 

1—378 (6) 


The approximation in equation (6) is made on the 
assumption that (/n -|- 1) ~ /*,. If the numerator 
and denominator of equation (4) are divided by 
(/*, 4- 1), we have 


From equation (11) we find that the equivalent 
generator of V, can be represented by an equiva- 
lent negative resistance 


~ Ma (Rpa + Z;,) 
Ma + 1 


1—378 (12) 


It can be seen that p is smaller in magnitude than 
(Rp, -f- Zl) , BO that the total Vi branch resistance 
is positive. Defining the V> branch impedance to 
be Z„ we have 

Z, = p -j- Rp2 4- Zi, = 1-^78 (13) 

Ma + 1 


WADC TR 56-156 


247 



Section I 

Crycto! OKillators 


On substituting equations (7) and (13) into equation 
(10), we have 

Ip2 = 1-378 (14) 

1^2 


and 

Q /> _ El Ml Zit 

Egi Rpi + Zk 

it will be found that 


1—378 (23) 


so that equation (9) may be written 


Gi' G." Gg G* = G, G, G, = 


= ( 


Zj + R2 


\ 


r Ri Zl y 

)l. 

1—378 ( 15 ) 

[.(Z2 4 " R2) R "b R2 Z2 


Ml Zk 


Rpi + Zk 


On substituting equations (13), (14), and (15) into 

( 11 ), 

_ I2 R (Z2 + R2) , I2R2Z2 


1-878(16) 


1—378 (24) 

Since Rpi and R,Zi, are very large compared with 
Zk and [(Z, + R,) R + Rs Z,], respectively, we 
can simplify equations (21) and (24) by writing 


R2 Zl Zk gml 

(Z2 + R2) R + R2 Z2 


1—378 (26) 


Thus 

G2 


Eg2 R2 Z2 

El R (Z2 ”i" R2) "b R2 Z2 


1—378 (17) 


To find G3 (= Ep/Eg2), we see that 


On dividing both numerator and denominator by 
(Z, 4- R,) and substituting for the values of Zgj 
and Zf as defined in equations (6) and (7) of fig- 
ure 1-173, equation (25) can be simplified some- 
what. Thus 


Ep = Ip2 Zl 1—378 (18) 

which, on substituting the value of Ip2 given in 
equation (14), becomes 

Ep = 1—378 (19) 

Dividing by Eg2 (= I2 R2). we have 

G3 = 1^ = 1-378 (20) 

J!,g2 02 

The conditions for equilibrium as expressed by 
equation (1) are thus found to be 


Gi G2 G3 — 

Ml Zk R 2 Zl _ - 

[Rpi -h Zk (mi + 1)1 [(Z2 4- R2) R 4- R2 Z2] 

1—378 (21) 


By a slightly different approach, in which the 
equilibrium is expressed as 


Gi' Gi" G2 G3 


where 


Egi El Eg2 _ 

Ep Egi El Eg2 


1 


Zg2 Zl Zk gml ^ j 1—378 (26) 

Z2Z, 

1-379. The design of a Butler oscillator must be 
such that under no-signal conditions the left side 
of equation 1 — 378(26) is greater than unity. As 
oscillations build up, the principal effects will be 
a decrease in the effective gm, and go,, as the sig- 
nal swings farther into the lower bend of the EcR 
curve. How large the equilibrium amplitude will 
be depends upon how much greater than unity the 
left side of equation 1 — 378(26) is at the Start. 
The larger the left-side magnitude, the greater 
must be the decrease in g,„,, and hence the greater 
the equilibrium activity must be. If the oscillator 
is to operate class A, as is usual, the gain equi- 
librium should very nearly hold for no-signal con- 
ditions, with due allowance made for a maximum 
Z, (= R 4- Zgj) when R = R„, the maximum series 
resistance permissible for the particular type of 
crystal unit chosen. With all else constant, maxi- 
mum activity is to be obtained when g^, and gm, 
are maximum under no-signal conditions. Assur- 
ance that the crystal unit will not be driven beyond 
its rated power can be approximately predicted 
from the plate characteristics of the tube to be 
used. If no grid current is drawn, the bias on Vi 
will be 




R2 Zl — (Z2 R2) R — R3 Z2 


R2 Zl 


1—378 (22) 


Ep, = - Ib, Ri 1-379 a) 

where Ipi is the average d-c plate current of 


WADC TR 56-156 


248 



Vi. Grid current can be drawn if [(niax)Ep — 
(max)E,] is greater than IbiRi, in which case the 
bias will be 

Eoi = (max) El — (max) Ep 1 — 379 (2) 

Greater amplitude stability is achieved if R, is 
sufficiently small for equation (2) to apply. Using 
the appropriate equations in paragraph 1-378, a 
maximum value of E, can be determined that will 
not allo w the crystal current, Ic, to become greater 
than V^cm/R. where Pen. is the maximum recom- 
mended power level of the crystal, and R is any 
crystal resistance between Rm and Rm/9. Very pos- 
sibly, a twin triode may be preferred, or perhaps 
the choice of tubes will be dictated by the h-f type 
of tube most readily available. Rj and Ra are to be 
kept as small as possible in the interest of fre- 
quency stability. In the final analysis, the plate 
voltage permitting an optimum output for the 
average crystal unit, without risking an overdrive 
for any expected value of crystal R, is most easily 
checked by experiment. 

DESIGN CONSIDERATIONS TO MAXIMIZE 
THE FREQUENCY STABILITY OF THE 
BUTLER OSCILLATOR 

1-380. If the cathode follower operates into a 
purely resistive network, as is indicated in the 
equivalent circuit of figure 1-173 (B), maximum 
stability in the phase characteristics is obtained. 
As nearly a resistive circuit as possible is desir- 
able, for under these conditions the frequency is 
independent of the plate resistance of the tubes, 
and a small increment of reactance requires the 
least adjustment of the crystal to restore a phase 
equilibrium. E., the output voltage of the cathode 
follower, is in phase with the excitation voltage, 
Ep. In a resistive circuit, Ep will be 180 degrees 
out of phase with E, 2 . Thus, Ep. must be 180 de- 
grees out of phase with E,. Since Egj is the voltage 
of ground with reference to the cathode of Va, and 
El is the voltage of the cathode of Vj with refer- 
ence to ground, U and U must be in phase. For 
example, imagine that R is zero, then Ri and Ra 
could be assumed to be two halves of a single re- 
sistance. In this case Ega would equal — E,, and 
the proper phase relation would exist. 

1-381. To maintain as nearly as possible a resis- 
tive circuit, Zp must tune as broadly as is practi- 
cable : the tendency of the input capacitances of V, 
and Va to shift the phase must be compensated ; the 
transit time of the vacuum tubes must be mini- 
mized ; and, if an inductor is connected across the 
crystal unit to antiresonate with the shunt capaci- 

WADC TR 56-156 


Section I 
Crystal Oscillators 

tance, C„, the resulting parallel-resonant circuit 
must also tune very broadly. The tuned plate cir- 
cuit, Zp, must be sufficiently selective to ensure 
that the circuit can oscillate only at the desired 
harmonic of the crystal frequency, but, beyond 
this, any increase in the tank selectivity only re- 
sults in a greater phase shift, and consequently a 
greater frequency shift, for a given percentage 
change in the plate capacitance. The use of a 
damping resistance as indicated in figure 1-173 (A) , 
or a low-Q coil, will broaden the tuning of the 
tank. The stray capacitance from the plate of Vj 
to ground should be kept to a minimum. 

1-382. To annul the input capacitance of Vj, which 
is equal to the total capacitance between the cath- 
ode of Vj and ground, we can connect an inductor 
in series with Rj, or replace Rj with a low-Q in- 
ductor and employ gridleak bias for Vj (while 
keeping the grid at r-f ground by the use of an r-f 
bypass capacitor), or shunt Rj with an inductor 
in series with an r-f bypass capacitor. In any event, 
the inductor is to be antiresonant with the cathode- 
to-ground capacitance at the operating frequency. 
With Ra acting as a damping resistance, a broad- 
band response is ensured for the antiresonant 
combination. 

1-383. The grid-to-cathode capacitance and the 
cathode-to-ground capacitance of V, can also be 
annulled by the use of antiresonant inductors. 
However, a more effective and economical method 
is to design the circuit so that the two cathode 
capacitances of V. neutralize each other regardless 
of the particular frequency. The grid-to-cathode 
capacitance, Cgd, is illustrated by the dotted-line 
circuit in figure 1-173 (B). The voltage across Cgci 
is Eg., so that the leading component of current 
through the grid circuit is 



For convenience, let it be imagined that all of Igg 
flows through R and Rpj in completing its circuit. 
If it is not to upset the phases of the voltages 
across these resistances, Igx must be annulled by 
an equal lagging current through the R-Rpj-Zp 
circuit. Thus, assuming the transit-time effect is 
negligible, the plate tank must be slightly induc- 
tive if Vj is to operate into a purely resistive load. 
This much can be controlled by the adjustment of 
the Vj plate circuit. With the circuit properly ad- 
justed, it can now be imagined that Igj is no longer 
a part of !„ but circulates directly through Zl and 
Cp,i in series. The design problem is to ensure that 
no part of I,, flows through Ri or V„ but returns 


249 



Section I 

Crystal Oscillators 

to Zl by flowing entirely through the Vi cathode- 
to-ground capacitance, C, — not shown in figure 
1-173. Furthermore, the design should be such that 
1,1 is all the current that flows through Ci ; other- 
wise, there will be a net unneutralized leading com- 
ponent upsetting the voltage phases in the rest of 
the circuit. With proper neutralization, the only 
reactive current will be confined to a series circuit 
comprised of Cgci, C„ ground, and an effective in- 
ductance shunting Zj.. The voltage across each of 
the reactive impedances due to I,, will be equal in 
magnitude and phase to the voltages caused by the 
in-phase currents flowing through the correspond- 
ing resistive impedances. Thus, to neutralize the 
circuit, the leading current, I,,, flowing through 
C, must of itself produce the voltage E,. This 
occurs when 


El — Ig, Xci 1 — 383 (2) 

or, using equation (1) to replace I,,, 

El = 1_383 (3) 

Xogl 

Now, 

El = = gnu Eg, Z, 1-283 (4) 

Using equation (4) to eliminate E, in equation (3), 
we have 

1, E Z Xci 

gml ^*1 Ok — = 

or 

g™. Zk = ^ 1-383 (5) 

'-'I 


Equation (5) defines the ratio for the V, input 
to output capacitance that should exist for maxi- 
mum frequency stability. 

1-384. To minimize the tendency of the transit 
time to cause the respective plate currents of V, 
and V, to lag the equivalent generator voltages, 
small-dimensioned h-f tubes should be used, and 
the plate voltages should be as high as practicable. 
If the additional expenditure in the design and 
production of the circuit are warranted, suitable 
networks can be devised to neutralize the transit 
effects. 

1-385. At the higher frequencies the series resist- 
ance of the crystal unit tends to increase, since 
the lagging component of current through the 
series arm must increase in order to annul the 


increased leading component through the shunt 
capacitance, C,. For this to occur, the frequency 
may need to be increased considerably above the 
natural resonance of the motional arm, so that the 
effective series resistance approaches the value of 
a parallel-resonant impedance. To reduce the re- 
sistance to the series-arm value, €„ should be 
annulled by an antiresonant inductor having an 
inductance 

L„ = 1-385 (1) 

w Co 

To prevent the crystal shunt reactances from be- 
ing more frequency sensitive than the plate circuit 
of V 2 , a shunt resistance should also be connected 
across the crystal unit to dampen the LoC, tank. 
A suitable resistance, Ro, that can interfere very 
little with the crystal stabilizing effect is 

R. = 6 R™ 1—386 (2) 

where Ro. is the rated maximum permissible crys- 
tal series resistance. 

STABILIZING EFFECT OF CRYSTAL 
IN BUTLER CIRCUIT 

1-386. From equation 1 — 241(2) we found that 
for a crystal operating at series resonance the 
fractional change in frequency required to produce 
a small change in phase, is expressed by 

dtp _ Ro 

(i> dfl 2-\/L/C 

where Ro is the total resistance the crystal faces, 
including the crystal’s own resistance, and L and 
C are the series-arm parameters of the crystal 
unit. In the Butler circuit (refer to figure 1-173) 
the crystal operates into a resistance 


R. = Z, + 1-386 (1) 

where Z, is the resistance of the feed-back circuit, 
R Z 

and p V. is the output resistance of Vi. On 

Ki "I 

substituting the values for Z, and Zt, we have 


R. - R + 


Rj (ais + 1) + Rps + Zl 


+ 


Ri Rpi 


1—386 (2) 


Ri (mi + 1) + Rpi 

If we assume that Rp, = Rp 2 = Rp > > Zl, and that 


WAOC TR 56-156 


250 



Pi = Ma = M > > 1» equation (2) becomes 


R, = R 4- 


RaP + Rp RiP "I" Rp 


or 


Ro = R + 


Ra 


+ 


Ri 


Rag. + 1 Rig. + 1 


1—386 (3) 


From equation (3) it is seen that the Q of the 
crystal circuit, and hence the frequency stability, 
is to be improved if Rj and R, are kept as small 
as possible and the transconductance of each tube 
is high. If R, and R, are of such values that the 
denominators in equation (3) are large compared 
with 1, a limiting value is approached, where 


(max) R„ = R -f — 1—386 (4) 

g. 

DESIGN PROCEDURE FOR 
BUTLER OSCILLATOR 

1-387. In considering the use of a Butler oscillator 
for controlling frequencies below 20 me, the prin- 
cipal factor to consider is whether the frequency 
stability required is greater than that which is 
normally obtained with a Pierce circuit. If not, 
there is little to gain by using two tubes and a 
frequency-sensitive tuned circuit, unless it is very 
important that the waveform in the output be 
more nearly sinusoidal and less influenced by the 
variations in the crystal resistance. The frequency 
stability of an average Butler, circuit can be ex- 
pected to be approximately 0.0005 per cent as com- 
pared with a stability of approximately 0.001 to 
0.0015 per cent for an average Pierce circuit. 
Above 20 me, the principal competitor of the But- 
ler is the transformer-coupled type of oscillator. 
The chief advantage of the Butler is its relative 
ease of adjustment and dependability. A border- 
line replacement crystal unit or an aging crys- 
tal unit, as a general rule, is more likely to be 
operative in the two-stage, cathode-coupled circuit 
than in any of the other types of v-h-f oscillators. 
Once the Butler circuit has been selected as the 
most appropriate to use, a crystal unit that has 
been series-tested at the intended frequency should 
be selected. The required minimum frequency tol- 
erance and the operating conditions to be expected 
determine whether the crystal unit, or perhaps the 
entire oscillator, is to be oven-controlled. For the 
next step, it is probably best to select the types 
of vacuum tubes to use. Insofar as space, weight. 


Section I 
Crystal OKillaters 

and cost are concerned, a single tube envelope for 
both amplifler stages is desirable. On the other 
hand, it may be found that a more balanced ar- 
rangement and more direct circuit connections can 
be had with separate tubes, particularly if the 
crystal unit is to be oven-mounted. The transcon- 
ductance and plate resistance of the tubes should 
both be high, for maximum frequency stability. 
For the same amplification factor, the tube with 
the larger g„, is usually to be preferred. For h-f 
and v-h-f operation miniature tubes are prefer- 
able, in order to reduce the transit time and the 
electrode-to-ground capacitances. For class-A op- 
eration, both tubes can be of the same type. For 
class-C operation, the power rating of the cathode 
follower should be greater than that of the 
grounded-grid amplifier. For maximum stability 
it may be desirable to isolate the load from the 
rest of the circuit, or to tune the load circuit for 
frequency multiplication. In this case a pentode 
can be used for either the cathode follower (usu- 
ally) or for the grounded-grid amplifier, with the 
load taken from the electron-coupled plate circuit, 
and with the screen grid serving as the oscillator 
plate. Either pentodes or triodes can be used in 
the basic Butler circuit, as desired. In the v-h-f 
range, triodes have the advantage of smaller tran- 
sit-time effects. Assume that it is intended to oper- 
ate the tubes class A. To reduce the transit time, 
to increase g„„ and to permit a minimum value of 
Zl (low-Q tank), the plate voltages should be as 
high as practicable. Determine the values of R, and 
R, that will provide a normal cathode bias for 
class-A operation. With all else equal, the feed- 
back transmission losses are a minimum if Ri = Rj. 
For class-C operation, Rj should equal approxi- 
mately 4Ri. Assume that R of the crystal unit is 
the maximum permissible value, and that the ef- 
fective gni’s of the vacuum tubes are 25 per cent 
less than their rated values for class-A operation 
at the selected plate and grid voltages. With these 
assumptions, determine the value of Zl that is re- 
quired to make the gain equation, 1 — 378 (26), 
hold. Rg should be large compared with Zp, so that 
Zl Zp. The plate tank represented by Zp can be 
designed as a high-Q circuit, antiresonant at the 
operating frequency, and shunted by a simulated 
load resistance, Rl, much smaller than the anti- 
resonant impedance of the tank, itself. In this case, 
Zl Zp =« Rl. The approximations above will be 
sufficient to build an experimental circuit that 
should oscillate in a free-running state. A variable 
resistance can be connected to simulate a crystal 
unit at series resonance. By varying the simulated 
crystal resistance over the range possible for a ran- 


WADC TR 56-156 


251 



section i 

Cryctai Oidllatert 

dom selection of crystal units, Ri, in the plate cir- 
cuit can be adjusted, if necessary, to ensure ttiat 
oscillations occur at ail possible values of crystal 
resistance without driving the crystal at a higher 
than recommended level. The empirical optimum 
value of Ri. can be accepted as the value of Zl to 
achieve in the design of the output circuit of the 
grounded-grid amplifier. The actual design of the 
output stage depends, of course, upon the type of 
load into which the oscillator is to operate. The im- 
portant consideration is that an effective resistance 
having the value of the experimental Rl is to be 
introduced in one way or another across the plate 
tank. The final problem is to neutralize the various 
circuit capacitances. In neutralizing the V, cathode 
capacitances, the adjustment which permits the 
feed-back circuit to be purely resistive can be ex- 
pected to coincide with the conditions for maxi- 
mum output amplitude and maximum crystal cur- 
rent. The design procedure discussed above should 
be accepted simply as a suggestion. Individual 


engineers may well prefer that primary attention 
be given to fitting the design to meet special 
requirements. 

MODIFICATIONS OF THE BUTLER 
OSCILLATOR 

1-388. As in the case of other conventional oscil- 
lator designs, the number of modifications of the 
Butler circuit appear to be unlimited. In figure 
1-174, the basic electron-coupled circuit is shown 
in (A), and a circuit employing a common ground 
return for the two tubes is shown in (B). In the 
electron-coupled circuit, the load, represented by 
Rl', is effectively isolated from the oscillator cir- 
cuit, in which the screen of Vi serves as the 
cathode-follower anode. The plate circuit of V, can 
be tuned to the second or third harmonic of the 
oscillator frequency, if desired, in which case Vi 
should be operated class C. For maximum output 
voltage, the plate impedance (rf Vi should be high. 
In figure 1-174 (B), the low-Q inductor, Li, is 




flgurm 1-174. (A) ffectroii-coup/ed Buffer clniHt. (B) Buthr drcuH having common cathodn ground roturn 


WADC TR 56-156 


252 



antiresonant with the distributed capacitance, C„ 
as is L. with Ce. Gridleak bias is employed with 
both tubes. However, the grid of V, is still kept at 
r-f ground through the bypass capacitance. It can 
be seen that equation 1 — ^78 (26), 

Ztf Zt, Zi, gmi ^ . 

Z,z, 

when applied to the common-ground return cir- 
cuit, becomes 

= 1 1—388 (1) 

Zj 

since Z* is equal to Z,. Also, since Z, = «« 

— , since Z^ where Rj now represents 

gm> tit 

the antiresonant impedance of the L,C, combina- 
tion, it can be shown that equation (1) can be 
expressed as 

Zl g„.i - ^ 1 1-388 (2) 

Ra g,ia 

If the product Rigms is large compared with 1, at 
equilibrium gmi must approximately equal 1/Zi,. 
1-389. The schematics of a number, of Butler cir- 
cuits employed by the military services are shown 
in figure 1-176. Circuit (A) is a receiver hetero- 
dyne oscillator that can be switched from crystal 
to manual operation simply by shorting out the 
crystal. C, is an r-f bypass capacitor that prevents 
the inductor L, from shorting out the cathode bias 
developed across Ri. L, is designed to be antireso- 
nant with the grounded-grid amplifier cathode-to- 
ground capacitance. C, is inserted to resonate with 
the distributed inductance of the crystal leads and 
feed-back circuit. Cj is a split-stator capacitor 
which permits the use of a grounded rotor, tiiereby 
reducing intersectional capacitances. Since the r-f 
current through C, is small, wiping contacts can 
be used for grounding without introducing noise. 
C, is simply a trimmer which permits adjustment 
of the effective inductance of L,. The plate side of 
C, effectively has a relatively large fixed compo- 
nent due to the stray plate-to-ground capacitance. 
To keep the plate tank balanced, an equal amount 
of fixed capacitance, Cg, is added to the other side 
of Cg. Rg is added to suppress parasitic oscillations. 
As can be seen, the output is obtained by split- 
load operation of the cathode follower. Since the 
output is delivered to a mixer circuit, where the 
effective load might be expected to undergo slight 


Sacrien I 
Crystal OsdNotors 

changes, it is probable that the loaded cathode- 
follower plate circuit is less frequency sensitive 
than the finely balanced tank in the V, plate cir- 
cuit. Since the balanced tank must be sufficiently 
selective to stabilize the frequency during manual 
operation, it cannot be loaded as would normally 
be done. Although the crystal is not active in the 
circuit during manual operation, it cannot be 
removed without resulting in an increase in fre- 
quency. This is due to the decrease in the cathode- 
to-ground capacitance that results when the crystal 
unit is removed. 

1-390. Figure 1-175 (B) (C) (D) (E) (F) is a 
composite arrangement of five different oscilla- 
tors, none of which have all the components shown. 
For example, the B+ return of V, is through R. 
in (B), (E), and possibly (F) ; it is through R, in 
(C), and also in (D), except that R, and R, are 
one and the same in the latter circuit, although the 
actual circuit is not indicated in the schematic 
shown. In circuit (E) the F, output is cathode- 
coupled to a mixer tube (6AK5W). An r-f choke 
is connected between the cathode and ground, not 
the resistance R,o. In circuit (B), Cg is actually 
composed of two 1.5-/v»f capacitors in series. The 
F, output is developed across a tuned tank identi- 
cal with and also inductively coupled to, the plate 
tank in the V, plate circuit. L, in circuit (B) thus 
serves as a transformer primary. The Fg output 
is fed to the grid of one and to the cathode of a 
second 6AG5 mixer stage. The heterodyned out- 
put of the first is 20 to 30 me, and that of the 
second is 4.8 to 5.7 me. 

1-391. The circuit shown in figure 1-175 (G) is a 
carefully designed experimental model that was 
built and tested during an investigation of h-f and 
v-h-f oscillators by a research team headed by 
W. A. Edson at the Georgia Institute of Technol- 
ogy. A breadboard model of this circuit was oper- 
ative at frequencies as high as 150 me, with crystal 
resistances as high as 500 ohms. Circuit (G) was 
found to have a frequency stability of 0.22 parts 
per million per volt change in the high-voltage sup- 
ply. The frequency was controlled at 126 me, the 
ninth harmonic of a 14-mc fundamental. The shunt 
capacitance of the test crystal was 12 /i/if, and the 
series resistance after tuning out the capacitance 
was 300 ohms. Oscillations could not be sustained 
at plate voltages below 60 volts, and the frequency 
instability increased greatly at voltages above 85 
volts. Note that circuit (K) in figure 1-175, which 
has been designed to operate with low- and me- 
dium-frequency, fundamental-mode crystal units, 
is not a true Butler circuit in that neither tube 
is operated as a cathode follower. 


WADC TR 56-156 


253 


Section I 

Crystal Oscillators 


F» or 



figure 1-175. ModWeatioas of Butler oscillator. Dotted lines In circuit (G) indicate stray capacitances 


Fig. 

Equipment 

Purpose 

Fi 

Fj 

F, 

Ft 

CR 

Ri 

Rs 

Ri 

Rs 

Rs 

Rs 

Rr 

Re 

(A) 

Radio 

Receiver 

R-266/ 

URR-18 

Hetero- 

dyne 

oscillator 

20.3- 

34.9 

2Fi 

■ 

■ 

CR-24/U 

(5th 

mode) 

0.18 

0.01 

6.8 

0.18 

12 

6.8 

■ 

1 

(B) 

Receiver 

Trans- 

mitter 

RT-178/ 

ARC-27 

3rd 
trans- 
mitter 
osc or 

2nd re- 
ceiver 
osc for 
hetero- 
dyning 

25.7- 
34 7 

F, 

T 

NA 

CR-23/U 

1 

1 

100 

1 

0.16 

NA 

12 

NA 

(C) 

Receiver- 

Trans- 

mitter 

RT-178/ 

ARC-27 

1st 

guard 

receiver 

local 

osc 

37.266 


NA 

2Fi 

CR-2S/U 

0.39 

0.12 

47 


0.22 

0 12 

6.8 

1 

(D) 

Receiver- 

Trans- 

mitter 

RT-173/ 

ARC-33 

1st mon- 
itor osc 
of trans- 
mitter 
M.O. 

0.8333 

F, 

4F,- 

18Fi 

Fi 

CR-28/U 

8.8 

S3 

100 

1 

1 

Same 

re- 

sistor 

as 

Rs 

1 

1 ^^ 


Receiver- 

Trans- 

mitter 

RT-178/ 

ARC-83 

Guard- 

channel 

hetero- 

dyne 

oscill- 

ator- 

doubler 

65 . 668- 
58.169 

F. 

NA 

2Fx 

and 

4F, 

CR-32/U 

0.12 

8 2 

38 

1 

0.33 

NA 

27 

00 

(F) 

Radio 

Receiver 

R-252A/ 

ARN-14 

Ist 

heter- 

odyne 

oscillator 

44.276- 

57.276 

F, 

NA 

2F, 

CR-23/U 

■ 

8.2 

■ 

0.1 

1 

7 

00 

■ 


Circuit Data for Figure 1-176. F in me. R in kilohma. C in /i/il. L in ah- NA (not applicable) means that no connections of 
any kind exist between points indicated. Question mark (7) indicates that schematic of the associated part of the circuit is not available. 


WAOC TR 56-156 254 































































































Section I 
Crystal Oscillators 


cs 



figurm I- 175. Contlnu 0 d 


R« 

Rio 

o 

C2 

Ca 

Cl 

Cs 

lO 

Cr 

Cs 

C, 

L, 

u 

La 

Li 

La 

V, Vj 


■ 


200 

14.3- 

43.6 

0.5- 

3.0 

15 

1000 

1000 

■ 

1 

1.2 

■ 

■ 

■ 

■ 

GL6670 

1 

NA 

1-8 

1 

1500 

47 

0 75 

NA 

NA 

NA 

NA 

00 


00 

NA 

1 

12AT7 

NA 

27 

0 

20 

3000 

100 

NA 

24 

500 

20 

0 

00 

00 

00 

■ 

■ 

12AT7 

NA 

470 

0 

104 

■ 

470 

470 

100 

25. 

000 

170 

0 

00 

CO 

00 

■ 

1 

6670 

8.2 

Cath- 

ode 

of 

mixer 

1 

5 

1 

10 

NA 

1.6- 

5 

00 

10 

10 

00 

CD 

1 

1 

1 

5670 

? 

■ 

i 

12 

2000 

100 

NA 


♦ 

? 

? 




? 

■ 

12AT7 


WAOC TR 56-156 


355 
























































































^2,000 ^jpo 


Fig. 

Equipment 

(G) 

Experi- 

mental 

osc 

(H) 

Radio 

Receiver 

R-540/ 

ARN-14C 

(I) 

Radio Set 
AN /ARC- 
34 (XA-1) 

(J) 

Radio Set 
AN/ARN- 
21(XN-2) 


44.275- 

57.276 

(14 

crystals) 


55.67- 2F, 

68 . 17 and 
4F, 


CR 

R. 

R. 

R. 

9th har- 
monic; 

0 05 

0 05 

5 

series 

resist- 




ance 




equal to 
Rs 




CR-23/U 

100 

8.2 

8.2 

CR-32/U 

0 22 

0.66 

33 

CR-23/U 

0.22 

■ 

10 


R, R, 


osc and 
multiplier 


Circuit Data for Figure 1-175. F ir me. R in kilohma. C in «*(• L in iih. NA (not appiicable) means that no connections of 
any kind exist between points indicated. Question mark (?) indicates that schematic of the associated part of the circuit is not available. 


WADC TR 56-156 

























































WAOC TR 56-156 


257 



































Section I 

Crystal Oscillators 


Fig. 

Equipment 

Purpose 

a 

D 

D 

a 

CR 

R, 

Q 

R, 

R. 

R» 

R. 

R, 

R. 

(K) 

Signal 

Generator 

SG- 

34(XA)/ 

UP 

Lf-F and 
m-f osc 

0.10 

0 18 

1 75 

1 85 
1.90 
1.95 

1 

1 

1 

CR-16/U 

(LF.) 

CR-19/U 

(HP,) 

10 

15 

0.027 

560 

39 

1000 

0.27 

1 

(L) 

Signal 

Generator 

SG-13/ 

ARN 

200-mc 

generator 

for mixing 

with 

lower 

freq. 

signals 

60 

4Fi 

1 

1 

CR-23/U 

58 

3.3 

0.015 

0.01 

0.27 

0.01 

0.27 

1 


Circuit Data for Figure 1 — 175. F in me. R in kilohma. C in L in ah- NA (not applicable) means that no connections of 

any kind exist between points indicated. Question mark (?) indicates that schematic of the associated part of the circuit is not available. 


SAKSW 




Ffffvre 1-175. ConUnumd 


(L) 


Transformer-Coupled Oscillator 

1-392. At the present time, the transformer- 
coupled crystal oscillator (see figfure 1-176) is not 
being widely used. It was during the v-h-f oscilla- 
tor investigation at the Georgia Institute of Tech- 
nology for the Signal Corps in 1950, mentioned in 
the last paragraph, that the transformer-coupled 
oscillator appeared to be the most promising for 
all-around versatility and general-purpose use. 
First, there is the advantage of a single-tube oscil- 
lator. Secondly, for low-power output (about four 
times the crystal power), the frequency stability 
has been found to be slightly superior to that of 
the average Butler circuit. Thirdly, with properly 
desigfned phase-compensating networks, an un- 
tuned pass band of 10 me is possible. Finally, with 
a relatively small sacrifice in frequency stability, 
the circuit design can be such that the power out- 
put is increased several fold without exceeding the 
recommended maximum drive level of the crystal 
unit. Although the transformer-coupled oscillator 
can perform satisfactorily at lower frequencies, 


its chief application is for control and generation 
of harmonic-mode frequencies above 20 me. The 
oscillator is generally designed for class-C opera- 
tion. A significant disadvantage is that the circuit' 
design for optimum performance characteristics 
— characteristics that can be approximately dupli- 
cated from one oscillator to another of similar de- 
sign — is generally more difficult to achieve than 
in other oscillator circuits. This is due chiefly to 
the difficulty in predicting the effective input im- 
pedance of tubes operated class C at frequencies 
where transit-time and stray-capacitance effects 
become appreciable. As a result, the theoretical 
and actual equilibrium conditions frequently are 
found to differ to a greater degree than in the 
average series-mode oscillator. More cut-and-try 
experimentation may prove necessary than would 
otherwise be the case. The operating principle of 
the grounded-cathode, transformer-coupled oscil- 
lator is closely allied to that of the grounded-grid 
and grounded-plate versions. The discussion and 
equations for the transformer-coupled oscillators 


WADC TR 56-156 


258 







































Section I 
Crytfal OscillcMon 


R. 

Rio 

D 

D 

a 

B 

B 

B 

B 

B 

B 

D 

B 

B 

B 

Li 

V, V, 


1 

1200 

10,000 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

12AU7 

1 

1 

500 

1.7-8. 7 

500 

100 

12 

500 

1 

1 

1 

1 

1 

1 

1 

1 

12AT7 


are, for the most part, based upon the analysis by 
W. A. Edson. 

PHASE CONSIDERATIONS OF 
TRANSFORMER-COUPLED OSCILLATOR 

1-393. Referring to figure 1-176, the useful load, 
represented by Rl, is connected across the second- 
ary of the plate transformer. The chief function 
of R, is to improve the frequency stability by low- 
ering the resistance of the crystal circuit, and to 
improve the amplitude stability by reducing the 
effect of variations in the input resistance of the 
tube. The parameter a is simply the constant of 
proportionality relating Ri. to R,. C, and C, are 
capacitors for tuning out the leakage inductance 
of the plate and grid transformers, respectively. 
The leakage inductance is equal to the high-side 
inductance multiplied by (1 — k’), where k is the 
coefficient of coupling. It can be directly measured 
at the low side of the transformer when the high 
side is shorted. Both transformers can be, simply, 
tapped coils. The crystal impedance is assumed to 
be the series-resonance impedance, R. The effec- 
tive turns ratios, N,, and N„ can be so chosen that 
Cp, and C„ annul each other’s effects. With the 


circuit properly designed, the tube operates into a 
resistive load, C,, being antiresonant with the 
dampened coil L„. I., I,., E,„ and E. (the voltage 

across the crystal) are in phase with E,,. The grid 
transformer thus provides the required 180-de- 
gree phase shift between E,, and E,. 

1-394. Where the resistance of the crystal series 
arm is not small compared with the shunt react- 
ance, Xco, the effects of C„ can be annulled by the 
conventional method of connecting an inductor 
across the crystal unit, by means of mutual induct- 
ance between the plate and grid transformers (to 
be discussed in connection with the grounded-grid 
oscillator), or by balancing the effects of C, 
against those of C„t. When the circuit is properly 
balanced, the crystal unit operates at the resonant 
frequency of the series arm. When C„ is balanced 
against Cp^, the leading component, I„, of the cur- 
rent through the crystal — that part through C„ — 
passes in its entirety, through Cj and the primary 
of the grid transformer. Similarly, the current 
through C,„ Ipp, passes in its entirety, through the 
secondary of the grid transformer. For this to 
occur, the voltages induced by the two currents in 
each section of the grid transformer must exactly 




figun 1-176. (B) Equivo/ant paralM- 
rasenant picrta drcuH wh»n crystal capac- 
itanea, C„ Is balancad by plats-to-grid 
FIgurm 1-176. (AJ Basic circuit of transformor-couplod oscillator capacitanco, 


WADC TR 56-156 


259 



















Section I 

Cryctol Occillaion 


annul each other. The transformer then appears as 
a short circuit to both currents. As indicated in fig- 
ure 1-176 (B), C„ becomes equivalent to a capaci- 
tor shunting the ground-connected half of Lp, 
thereby effectively increasing L,,, and C,„ becomes 
equivalent to an increase in C,, equal to Cp*. The 
balanced state is reached when 


I = N I 

■^O *pg 


or 


wCqEl — NgtoCpg Ep 


or 


C 


PC 


Co ^ 

Ng ■ Ep 


Co 

NgNp 


1—394 (1) 
1—394 (2) 

1—394 (3) 


Equation (3) can generally be realized with prac- 
tical values of Ng and Np if Cpg is on the order 
of 0.05 to 0.1 /ifif. Such values of Cp, can be ob- 
tained with screen-grid tubes but not with triodes 
unless a d-c blocked inductive arm is connected 
between plate and grid to annul most of the 
capacitance. 



(eff)R.E,^ 

Eg* 


(eff)Rg 

Ng* 


1—395 (6) 


where (eff) Rg is the effective grid resistance 
which takes into account the gridleak losses and 
the transit-time loading. For small values of Ng, 
the Rg' losses can be considered negligible. The 
plate power can be assumed to be. 


Pp = I. E. = 


El^ (Rl -1- R ■+• Ri) 
Rl (R -|- Rl) 


El* (R -|- Rl -t- a R,) 
a Rl (R Rl) 


and 


V ^ a Np* R, (R -h Rl) 
” Pp R H- R( -t- a R, 


1—395 (7) 


1-395 (8) 


on combining equations (2), (3), (4), (5), and (8), we 
find that at equilibrium 


Gi G2 G3 G4 


gp, a Np Ng Rl* 
R ■+• Rl (1 + a) 


1 


GAIN REQUIREMENTS OF TRANSFORMER- 
COUPLED OSCILLATOR 

1-395. Referring to figure 1-176 (A), it can be 
seen that at equilibrium. 


Gi G2 G3 G4 = ^ 


where 



El El ^ _ 2^ 

Ep El El 

1—395 (1) 


mZp 

Rp + Zp 


3m Zp 

1-395 (2) 


G2 = ^ = — 1—395 (3) 

Ep Np 



R. 

R -j- Rl 


1—396 (4) 


G4 = Ng 1—395 (5) 

Rg' in equation (4) is the effective resistance of 
the grid transformer as it appears in parallel with 
R,. It is equal to E,* divided by the grid losses. 
Thus, 


1—396 (9) 

Q DEGRADATION IN TRANSFORMER- 
COUPLED OSCILLATOR 

1-396. In Edson’s analyses of series-mode oscilla- 
tors, he employs a useful term which he calls the 
Q degradation of the crystal unit. It is defined 


D = ^ 1—396 (1) 

Iv 

where Rg is the total resistance which the crystal 
must operate into. In the transformer-coupled os- 
cillator, assuming that the transformer imped- 
ances that the crystal faces are large compared 
with Rl and R, of figure 1-176, 


D 


R -h Rl (1 a) 
R 


1—896 (2) 


As discussed in paragraph 1-241, the frequency 
stability of the series-resonant crystal unit is di- 
rectly proportional to the Q of the crystal circuit. 

Q 

It is therefore directly proportional to -g, where 

Q is the Q of the crystal, itself. Thus, with a given 
crystal unit, the frequency stability varies in- 
versely with D. If the minimum Q of a crystal unit 
is estimated from the maximum permissible C„, 
from the frequency, the harmonic, the particular 
crystal element, and the maximum permissible 


WADC TR 56-156 


260 



series resistance, and if the required frequency 
stability is known, then the maximum permissible 
D can be determined from the random phase shifts 
to be expected during: operation. In the averagre cir- 
cuit, it is sufficient, simply, to keep D as low as 
possible consistent with the output desired. When 
R, is expressed as a function of a, R, and D, the 
loop grain as defined by equation 1 — 395 (9) be- 
comes 


G, 


Gj G3 G4 


g„ a Np N, R (D - 1)^ 
D (a -f- D* 


’ Section I 
Crystal Oscillators 

Nk be as small as possible. For a griven value of 

the product N„N, can be a minimum when -, f -v, 

(see equation 1 — 396 (3) ) is a maximum. The 
maximum occurs when a = 1. With a, D, and g„, 
decided upon, equations (2) and (3) give mutually 
minimum values when N,, = N,. Thus, for broad- 
band operation, let 


Zp = Z, 


D + 1 
gn. (D - 1) 


1—398 (4) 


1^96 (3) 

LOAD-TO-CRYSTAL POWER RATIO OF 
TRANSFORMER-COUPLED OSCILLATOR 

1-397. The ratio of the load to the crystal power is 


p /p - ^1- _ (D + a)^ 

“ I,* R a (a + 1) (D - 1) 

1—397 (1) 

It is generally desired to have the power ratio as 
high as is consistent with satisfactory frequency 
stability. With a given value of D, the ratio be- 
comes large as a is made small. If the design is 
based on obtaining a griven minimum output with 
a given value of D when the crystal-unit resistance 
is a maximum (minimum D), the value of a can 
be determined by equation (1), and equation 
1 — 396 (3) can be used to determine the value of 
NpN, most likely to produce the required gm for 
the crystal to be driven at the desired level. 

BROAD-BAND CONSIDERATIONS IN THE 
TRANSFORMER-COUPLED OSCILLATOR 
1-398. For broad-band untuned operation it is im- 
portant to have Zp and Z, (the impedance faced by 
the grid) and the plate and grid capacitances as 
small as possible. Assuming that R„ is large com- 
pared with the resistance appearing across the 
secondary of the grid transformer. 


N,* R, (R -b a R) 
a Ri -f- Ri R 


or 

N, (D -H 1/a) 
‘ g„Np(D-l) 


Also, Zp can be expressed as 


1—398 (1) 


1—398 (2) 


Np (D + a) 

" gm N, (D - 1) 


1—398 (3) 


To keep Zp and Z( low, it is desirable that Np and 


WADC TR 56-1 5« 


261 


D should be as large as possible consistent with 
the required frequency stability. 

FREQUENCY STABILITY OF THE 
TRANSFORMER-COUPLED OSCILLATOR 

1-399. The frequency-stability equations for the 
transformer-coupled oscillator are 

and 

1-399 (2) 

where Q is the Q of the crystal. When equations 
1 — 398 (2) and (3) are multiplied by D, it can be 
shown the DZp is a minimum when 

" D = 1-1- VTTa 1—399 (3) 

and DZp is a minimum when 

D = l-h V 1 + 1/a 1—399 (4) 

For both to be a minimum simultaneously, a must 
be equal to 1, which means that 

D = 1+ VT = 2.414 1—399 (5) 

Under these conditions, the power ratio, as given 
by equation 1 — 397 (1), becomes 

^ = 4.12 1—399 (6) 

* C 

For larger power outputs, the value of a must be 
decreased and Np increased. Since the expected 
variations in the grid capacitance are generally 
larger than those in the plate capacitance, it is 
usually desirable to favor the grid circuit insofar 
as the frequency stability is concerned. 

1-400. The grreatest probability that the effect of 
a random vari4gtion in the grid capacitance will be 



Section I 

Cryttol Oecillator* 


canceled by a random variation in the plate capaci- 
tance occurs when the fractional change in fre- 
quency due to the average aC„ is equal to that due 
to the average aC,. If the average aQ is on the 
order of 10 times the average aC,„ then Zp should 
be equal to lOZg. Equations 1 — 399 (1) and (2) 
will then represent equal average variations in 
frequency. 


DESIGN PROCEDURE FOR TRANSFORMER- 
COUPLED OSCILLATOR 

1-401. The procedure to follow in designing a 
transformer-coupled oscillator depends upon the 
principal objectives to be sought in the design. 
That is, some fixed requirement serves as a start- 
ing point, and the design proceeds from there. One 
limitation that will be common to all the circuits 
is that the crystal power rating not be exceeded. 
This requirement, then, in the general case, can 
be the initial design consideration. The crystal 
power is 

P„ = R = 1-401 (1) 

Mg Ki 

Also 

p _ e,;r 

' ' (R, + R)^ Np“ (Ri + R)^ 

_ g,p^E/Zp^R 

Np^ (R + R.)^ 


or 


Pc 


E,^ a' R,^' Np“ 
R D" 


1—401 (2) 


Multiplying equation (1) by equation (2) and tak- 
ing the square root, we have 


e E ^ a N 

P„ = " 1-401 (3) 

D N,, 

Now, (gniE/) is assumed equal to (I, .Eg), which, 
in turn, is principally a function of the excitation 
voltage and the plate characteristics of the tube to 
be used. By equation 1 — 395 (9) (also by equating 
equation (1) to equation (2) ) 


3m 


Rc 

a R,“ Np Ng 


1-401 (4) 


Rc, remember, is equal to RD, the total resistance 
that the crystal unit operates into. From equation 
(4) it can be seen that if Rc(= R -f Ri(l + a) ) 
is large compared with the maximum R(= Rm), 
the equilibrium transconductance will be approxi- 
mately the sanqe for all values of R. On the other 


hand, if the minimum D is small, the value of gm 
at minimum R may be as much as one-half its value 
for R = Rm. Note that some change must occur in 
gm, and hence in Eg, if R varies and the rest of 
the parameters remain constant. To ensure class-C 
operation for all values of R, let the class-A value 
for g.„ equal twice the equilibrium g„i according to 
equation (4), with R,. assumed to be a maximum. 
With the circuit so designed, the amplitude of the 
oscillations will build up until the tube is cut off a 
fair proportion of each cycle. Even if class-A oper- 
ation is desired, a reasonable difference should be 
allowed between the rated transconductance of the 
tube and the estimated equilibrium value when R,. 
is maximum. This should be sufficient to allow for 
all expected tolerances in the plate characteristics 
and in the tuning of the oscillator circuit. The per- 
centage variations in R from one crystal unit to the 
next is not quite as great in the v-h-f crystals as 
in the lower-frequency elements, since it is more 
important that the maximum permissible resist- 
ance be kept as small as practicable. 

1-402. The ideal design would permit the percent- 
age variations in Eg* to exactly equal in magnitude 
the percentage variations in R. Under these con- 
ditions, the crystal power, as indicated in equation 
1 — 401 (1) would be the same for each crystal 
unit. To approach such a design, g,„, as a function 
of Eg, would have to be known for the particular 
tube. Such an analysis is beyond the present dis- 
cussion, but the method to be used would be quite 
similar to that employed in the analysis of the 
effects of different values of crystal resistance for 
the Pierce circuit. As a rule of thumb, the average 
crystal R in the v-h-f range can be assumed to 

equal If the transformer-coupled oscillator is 

designed to drive the crystal unit at 50 per cent of 
its maximum rated power for the average R, there 
is little danger that crystal units having other 
values of resistance will be overdriven. For broad- 
band operation the crystal power will be approxi- 
mately directly porportional to R. When D is small 
the crystal power increases as R decreases as long 
as the percentage increase in E„* is greater than 

R 

the percentage decrease in R. With R = and 
p 

Pc = — equation 1 — 401 (1) gives an assumed 
value for E,’, thus 

(ave) Eg* = Ng* 

Rm 

or 

(ave) Eg = Ng Ri V Pc,„/R,., 1—402 (1) 


WADC TR 56-156 


262 



Note that equation (1) also gives the value of E» 
which would exist if a crystal of maximum resist- 
ance were driven at its rated level. However, 
equation (1) is less important for determining 
than it is for determining N^. It is assumed 
that a class-C value of g„, has been agreed upon. 
An approximate value of E^ corresponding to the 
chosen g,„ is thus already determined. After the 
value of R, is decided upon, equation (1) can 
serve to determine N*. 

1-403. The design procedure followed so far can 
generally be applied to any transformer-coupled 
oscillator. A v-h-f pentode and a harmonic series- 
mode crystal unit are selected. Regulated screen 
and plate voltages for the tube are decided upon. 
An approximate class-C value of g„ and the cor- 
responding Eg are estimated, and average values 
of Pr and R are assumed. Once that O and a have 
been selected according to the particular require- 

N 

ments of the oscillator, the ratio can be deter- 
mined from equation 1 — 401 (3), and N„ from 

N 

equation 1 — 402 (1). Or, in case a particulars^ 

ratio is to be preferred, a and D can be determine 
with the aid of equation 1 — 401 (3). An alterna- 
tive approach, which is the one to be followed 
when using the table in paragraph 1-404, is to first 
determine optimum values for D, a, and N,,/Nj, 

R 

assume an average crystal R = and a crystal 

p 

power equal to-^, and use equation 1 — 401 (3) 

to determine the value of gmEg* (= I„Eg). The next 
problem is to determine what value of Eg will pro- 
duce the required value of I„Eg when the tube is 
operating into a plate impedance that is small rela- 
tive to the tube R„. The equations relating I„ to Eg 
in the analysis of the Pierce circuit, or rather the 
basic methods used to derive the equations, are 
applicable here if modified properly. Since some 
trial-and-error will be required regardless, it may 
well be preferable to determine the correct Eg em- 
pirically. The selected vacuum tube can be driven 
by an external generator having a variable output 
at a frequency near the actual frequency for which 
the oscillator is to be designed. The tube should 
operate into a smalT resistive impedance and the 
gridleak resistance should be the same as that to 
be used in the final design. Eg should be varied 
until the measured I,, is such that the product 
agrees with the value computed from equation 
1 — 401 (3). With Eg approximately known, Ng can 
be determined by means of equation 1 — 401 (1). 
1-404. The dimensionless equations listed below 
relate the various parameters. These equations, 


Section I 
Crystal Oscillators 

some of which have already been given, are also 
useful in determining the various circuit voltages 
and currents when Eg is known, or the require- 
ments thereof, and in comparing the characteris- 
tics of oscillators of different design. 


g... Zp = 


gm Zg 


Np“ g,.. R 
Ng* g... R 
R,/R = 


PL/g.n E* 


Pl/Pc = 



(D-f a) 

Ng 

(D - 1) 

N, 

(D + 1/a) 

N 

p (D - 1) 


N, D (a + D* 


Ng a (D - 1)' 


Ng D (a -H 1)=* 


N,. a (D - D* 

D ■ 

- 1 

a 1 

2 _ 

N., (D -1- 


Ng D (D - 1) 


(D -H a)* 


1-M04 (1) 
1—404 (2) 

1-404 (3) 
1—404 (4) 
1-404 (5) 


a (a -h 1) (D - 1) 


1—404 (6) 
1-404 (7) 


The following table, prepared by the v-h-f oscilla- 
tor research team at the Georgia Institute of Tech- 
nology, lists the quantitative relations that hold 
for five typical designs of the transformer-coupled 
oscillator. D, a, and the N,,/Ng ratio are predeter- 
mined to provide optimum or practical operating 
characteristics according to five different objec- 
tives. 

A. Symmetrical circuit design that yields mini- 
mum values of ZpD and ZgD. By equations 1 — 399 
(1) and (2), this design permits maximum fre- 
quency stability if the average variations in C„ 
and Cg are approximately equal. Low power out- 
put. Narrow bandwidth. 

B. Nonsymmetrical grid and plate impedances. 
Designed for optimum frequency stability when 
(ave) ACg = 10 (ave) aC,,. Low power output. 
Narrow bandwidth. 

C. Nonsymmetrical grid and plate impedances. 
Maximum frequency stability when (ave) ACg = 
10 (ave) aC,. and the power output is 10 times that 
in design B, but the stability is less than that in 
designs A and B. 

D. Symmetrical circuit. Broad-band untuned 
operation. Provides small values for Z„C,, and ZgCg. 
Tubes require large transconductance and small 
input and output capacities. Z„ and Zg are one- 
half those in design A. Average AC,, assumed 
equal to average aC*. Frequency stability below 


WAOC TR 56-156 


263 



S«ctton I 

Crystal Oscillators 

average. Power output low, but greater than that 
in designs A and B. 

E. Nonsymmetrical design except that Zp = Z,. 


MODIFICATIONS OF TRANSFORMER- 
COUPLED OSCILLATOR 
1-405. Four modifications of the basic trans- 
former-coupled oscillator are shown in figure 
1-177. Circuits (A), (B), and (C) are experi- 
mental models that were designed at the Georgia 
Institute of Technology, but not in accordance 
with the designs given in paragraph 1-404. Cir- 
cuit (D) is an oscillator in actual use that has 
been designed specifically to operate with Crystal 
Unit CR-24/U without driving the crystal beyond 
its Military-Standard level. Figure 1-177(A) is a 


Af is the difference between the operating fre- 
quency and the series-resonance frequency of the 
crystal unit as measured with Crystal Impedance 
Meter TS-683/TSM. Figure 1-177 (B) is a circuit 
intended for broad-band untuned operation with a 
center frequency at 63 me. Satisfactory operation 
was obtained on a crystal plug-in basis from 53 
to 73 me. The designers recommend that the phase- 


High power output — same as that in design C. 
Broad-band untuned operation. Frequency stabil- 
ity below average. 


low-power circuit intended to be operated within 
a band of ±2 per cent of 55 me. When tested 
with a number of crystal units having overtone 
frequencies between 50 and 60 me, the circuit, 
with Lp and L, adjusted to be antiresonant with 
C„ and Cp, respectively, at 55 me, showed the fol- 
lowing operating characteristics. Figure 1-177 (D) 
shows a slug-tuned transformer-coupled oscillator 
that employs a battery-operated subminiature 
tube. This oscillator is used in Radio Receiver- 
Transmitter RT-159A/URC-4. 


compensating network, which was selected with 
the help of chart I, page 446, "Network Analysis 
and Fe^back Amplifier Design,” Bode, be shifted 
from the grid to the plate circuit. Figure 1-177 (C) 
is a circuit designed to provide an output ap- 
proaching 1 watt at 50 me. Due largely to difii- 
culties in predicting the input resistance of the 
vacuum tube, the differences between the initial 


TRANSFORMER-COUPLED OSCILLATOR DESIGNS 


Parameter 

Value of Parameter 

Design A 

Design B 

Design C 

Design D 

Design E 

Zp/Zp 

1 

10 

10 

1 

1 

a 

1 

1 

0.0985 

1 

0.24 

D 

2.414 

2.414 

2.414 

10.65 

10.65 

Np/Np 

1 

vio 

7.07 

1 

1.166 


2.414 

7.63 

12.65 

1.207 

1.315 

gn.Zp 

2.414 

0.763 

1.255 

1.207 

1.315 

gmZpD 

5.83 

18.4 

30.3 

12.85 

14.10 

gmZpD 

5.83 

18.4 

30.3 

12.85 

14.0 

Np^gmR 

4.828 

15.25 

104.0 

0.457 

0.854 

Np‘g,pR 

4.828 

1.525 

2.10 

0.457 

0.63 

Pl/Po 

4.12 

4.12 

41.2 

7.05 

41.2 

R./R 

0.707 

0.707 

1.286 

4.825 

7.78 

Pl 

8mEp» 

1.706 

5.4 

11.9 

0.662 

1.08 


f 

(me) 

Harmonic 

Cp 

(M^f) 

R 

(ohms) 

Stability 

(ppm/volt) 

Af 

(kc) 

Pl 

(mw) 

50 

6 

5 

48 

0.6 

1.6 


54 

9 

6 

60 

0.18 

0.6 

80 

54.8 

7 

6 

80 

0.11 

0.0 

, 60 

68.3 

7 

8 

82 

0.626 


40 

60 

3 

18 

25 

0.66 


40 


WADC TR 56-156 


264 


























































































S«ction I 
Crystal Oscillators 




Figure 1-177. ModMcationt of traitsformor-touplod o$eillator for; (A) High stability, 
(B) UntuHod broad band. (C) large output. (D) Slug-tuned, narrow band 


design and the final adjustments (the latter shown 
in figure 1-177 (C)) were quite large. The operat- 
ing characteristics of the circuit are given below : 
Frequency stability = 0.3 ppm/volt 
D-C power input = 1.8 watts 


R-F power output = 0.6 watt 
Crystal power = 0.02 watt 
Efficiency = 33 per cent 
Pl/Pc = 30 
Grid bias = —19 volts 


WADC TR 56-156 


265 





Saetion i 

Crystal Osciliotere 
Greanded'Grld Oscillator 

1-406. The grounded-grid oscillator, or single-tube 
Butler oscillator (see figure 1-178) is very similar 
in principle to the transformer-coupled oscillator. 
In fact, it can be described as a transformeiv 
coupled oscillator in which the feedback is cathode- 
coupled to the input. The cathode coupling elimi- 
nates the need for phase reversal in either the 
plate or input transformer. As indicated in figure 
1-178, the input transformer can be reduced to an 
r-f choke. For the plate circuit, an autotrans- 
former permits a maximum coefficient of coupling. 
Either a triode or pentode can be used. Although 
operable at lower frequencies, the oscillator is 
usually employed in the v-h-f range. The low input 
impedance reduces the effect of variations in the 
cathode capacitances. Unless the tube is operated 
class C, the crystal operates into a relatively low- 
resistance circuit. The power output is lower than 
that obtainable with a transformer-coupled oscil- 
lator, but for small outputs the frequency stability 


is quite high and the load is more readily shielded 
from the input. This latter feature reduces the 
possibility that the oscillator will operate at fre- 
quencies other than that of the desired mode of 
the crystal. The shunt capacitance of the crystal 
unit should be compensated by a broad-band anti- 
resonant inductor, or by other means. The most 
dependable method for use over a wide range of 
frequencies is to introduce mutual inductance be- 
tween the input transformer or choke and the 
plate transformer. This can be attained by wind- 
ing the cathode inductor on the same form as the 
plate transfbrmer is wound. The correct coefficient 
of coupling between the input and plate inductors 
is the one that permits the crystal to vibrate at its 
true motional-arm resonance. For a theoretical 
discussion of this mode of capacitance compensa- 
tion, see Eldson et al. The grounded-grid oscillator 
is most advantageous to use when maximum com- 
pactness and simplicity are desired in a low -power, 
broad-band, untuned v-h-f oscillator. There is the 
very important additional advantage that the oscil- 







Figure 1-178. 


I 

(C) (D) (E) 

Basic grcuitd»d-gr1d otcilittfcr and equivalent circuit pngreteively simplified 


WADC TR 56-156 


266 



lator can be readily designed so that both the out- 
put amplitude and the frequency stability are vir- 
tually independent of the crystal resistance. 

ANALYSIS OF GROUNDED-GRID 
OSCILLATOR 

1-407. Referring to figure 1-178, it will be assumed 
that the plate circuit is tuned to the desired har- 
monic, series-resonance frequency of the crystal, 
that the resistance in the plate tank is negligible, 
that the grid current is negligible, that the auto- 
transformer coefficient of coupling is unity, that 
the cathode-to-ground capacitance is antiresonant 
with the cathode choke, and that the r-f current 
through the choke is negligible. Under these con- 
ditions the equivalent r-f circuit of the oscillator 
is that shown in figure 1-178 (B). All voltage sym- 
bols are treated as unsigned. The polarities shown 
correspond to the instantaneous polarities that 
hold during the positive alternations of the r-f 
grid voltage. Eg. The current arrows point in the 
instantaneous direction of the in-phase electron 
flow. The reactive component of the plate tank 
current is not represented, although in reality it 
is primarily the “flywheel” current that produces 
the voltages E, and Ej. This reactive current in 
the tank is that which flows through Cp and is 
equal to mCpEl. If Rl were reduced to zero, there 
would be no reactive current and the transformer 
would effectively short-circuit the crystal to the 
plate. Of course, oscillations could not exist under 
these conditions, if for no other reason than the 
fact that E, would be reduced to zero and E,, being 
simply the voltage across the crystal unit, would 
be displaced 180 degrees from the phase required 
for oscillations to be maintained. E, must be 
greater than E.. The important feature to remem- 
ber is that insofar as the resistive component of 
the current is concerned the action of the auto- 
transformer is the same as that of a conventional 
transformer when the primary and secondary cir- 
cuits are connected in parallel as shown in figure 
1-178 (B). Note, however, that the turns ratio, N, 
is defined as the ratio of the total turns to the 
turns comprising L,. 

1-408. The power fed to the transformer is simply 

Pl == II El = Ip Ej 1 — 408 (1) 

Now, 

El = E. -I- E, = E, -f 

N 

or 

El = 1-408 (2) 

N — 1 


So 


Section 1 
Crystal Oscillators 

1 

II = 

Ip E. (N - 1) Ip 

El ■ N 

1—408 (3) 

And since 

Ip = 

Il + Ii 

1—408 (4) 

We have 

Ip = 

(N - 1) Ip , p 

N 

1-^08 (5) 

or 

Ip = 

NI. 

1^08 (6) 

and 

Ir. = 

I. (N - 1) 

1—408 (7) 


The ratio of the output to the crystal power is 
Pl/P, = II* Rl/I,," R = = Zl/R 

1—408 (8) 

Note that for a given turns ratio, the power ratio 
is directly proportional to the resistance ratio. 
/N — IV 

The term Rl f — — 1 is simply the equivalent 

load resistance, Z,,, that the transformer presents 
to Ip. The total impedance across the vacuum tube 
is thus 

Zp = Rl y + R = Zl -f R 

1—408 (9) 

1-409. It is convenient to imagine that the ground 
connection is at the point G' in figure 1-178. That 
is, let G' be our point of reference. Insofar as the 
r-f circuit is concerned such a supposition requires 
no alteration in the currents and voltages involved, 
but it does simplify the visualization of the circuit 
charactistics. The supposition does not mean that 
there is no difference in the r-f potential between 
G' and the actual ground. With G' in figure 1-178 
(B) assumed to be the ground connection, it can 
be seen that the tube is effectively connected as a 
cathode follower except that no load is taken from 
the cathode circuit. The crystal R is the cathode 
resistance, and E, is the voltage input to the grid 
circuit. As discussed in the analysis of the two- 
tube Butler circuit and as illustrated in figure 
1-173, the cathode-follower type of circuit can be 
represented by an equivalent circuit in which the 
plate-circuit resistance, exclusive of the cathode 

resistance, is equal to — times the actual re- 
A* + 1 


WADC TR 56-156 


267 



Saction I 

Crystal Oscillators 

sistance. Also, the equivalent generator voltage is 

— times the conventional value that would be 
M 1 

assumed if the cathode resistance were not pres- 
ent. As applied to the grounded-grid oscillator, the 
equivalent circuit is that shown in figure 1-178 (C) . 
Care must be taken not to interpret the power 
supplied to the plate-circuit resistances as being 

reduced by a factor of — With u and Rp 

/* + 1 

assumed large compared with unity and Z,„ respec- 
tively, circuit (C) reduces to circuit (D). 

LOOP GAIN IN GROUNDED-GRID 
OSCILLATOR 

1-410. At equilibrium 


Gi G2 G3 G 


4 


where 


^2. . .B. . ..^ . ^ = 1 

R 2 1^0 

1-410 (1) 


G = = R 

' E 2 ■ Rp -h Zl + R (m + 1) J_ R 

Sm 

1—410 (2) 

G2 = E,/E„ = Z,yR = 1^10 (3) 


G3 = El/Ei = N/(N - 1) 


1—410 (4) 


G4 = Ej/El = 1/N 1—410 (5) 

Equation (2) simply represents the gain of a 
cathode follower. It can be derived from the equiv- 
alent circuits in figure 1-178 in a manner similar 
to the cathode follower gain derivation in the 
analysis of the two-tube Butler circuit. In com- 
bining equations (2), (3), (4), and (5), we have 


Gi G2 G3 G4 


(Rp -h R -f mR) ^ 1 
Ri. (N - 1) (M + 1 - N) 

1—410 (6) 


If the approximate value for G, can be assumed, 
the gain equation can be expressed as 


1 Rl (N - 1) x> _ Zc x> 
gp, N - 1 

1—410 (7) 

ACTIVITY CONSIDERATIONS IN 
GROUNDED-GRID OSCILLATOR 

1-411. From equation 1 — 410(7) we can predict 

WAOC TR 56-156 268 


the relative activity to be expected with different 
load and crystal resistances. When oscillations first 
start, the term l/g,„ is a minimum, 'fhe amplitude 
increases until g„, is reduced by the increase in 
grid bias to the value that makes equation 1 — 410 
(7) hold. The greater the difference between the 
initial, zero-bias value of gm and the equilibrium 
value, the greater will be the final amplitude. The 
initial g„ of the tube should therefore be as large 
as possible if a maximum output is desired. For a 
given crystal R, the output is increased by increas- 

N — 1 

ing Ri, and making the ratio — a maximum, 

which occurs when N = 2. Assuming that N = 2, 
i.e., that the transformer coil is center-tapped at 
the connection to the crystal, equation 1 — 410(7) 
becomes 


1/gm - R1./4 - R = Zl - R 1—411 (1) 

In equation 1 — 408 (8), it was found that the 
power ratio, to which we shall assign the symbol 

r^ = ^Y is equal to Zi,/R. On substituting the 
value rR for Zl in equation (1) we find that 

r = 1 -I 1-411 (2) 

gm R 

By equations (1) and (2), which hold only when 
N = 2, we see that increasing the amplitude by 
increasing Ri. results in simultaneously increasing 
the power ratio. Also, note that the minimum 
power ratio can be predetermined as a function 
of the class-A value of g„ and the maximum per- 
missible R of the crystal unit. If variations in 
R from crystal unit to crystal unit are not to 
have a large effect upon the output amplitude, 

ZL^=-^^must be large compared with the maxi- 
mum crystal R. In other words, the minimum 
power ratio should be as large as possible consist- 
ent with the requirements of frequency stability. 

CRYSTAL DRIVE-LEVEL CONSIDERATIONS 
IN GROUNDED-GRID OSCILLATOR 

1-412. If P,„, is the rated maximum power of the 
crystal unit, the maximum permissible I„ for a 
given R is 

(max) Ip = (max) E, g„ = V Pom/R 

1—412 (1) 

Consequently, 


(max) El = (max) Ip Z,. = Zl V Pcm/R 

1—412 (2) 



By equations 1 — 410 (4) and (5) 


, ^ t:, (max) E, N Zl N V P.m/R 

(max) E, = = — N_i 


and 


(max) E2 


1-412 (3) 

(max) El/N = 


Assuming that N = 2, we have 


1—412 (4) 


(max) E|. 


= (max) El = (max) E2 

= 

4 



1—412 (6) 


The maximum permissible excitation voltage of 
the tube with a center-tapped transformer is 


(max) E, = 


(max) I, 


£ — 


ga 


(max) E2 — (max) 




1-412 (6) 

IV 


Equation (6) is to be interpreted as giving the 
maximum permissible E* for a given value of R. 
The smaller the value of R, the larger will be the 
permissible value of E,. For v-h-f crystal units in 
which the shunt capacitance of the crystal is not 
compensated the minimum R encountered may be 
on the order of Rm/5, where R^ is the rated maxi- 
mum. However, with capacitance compensation all 
values of R will be less than R„, and the minimum 
may well be on the order of Rm/9 or less. The 
plate characteristics of the vacuum tube and the 
electrode voltages must be such that an excitation 
voltage equal to the maximum permissible Ej, as 
defined by equation (6) , does not cause an effective 
Ip greater than \/I’cm/R- With a sharp-cutoff, grid- 
leak-biased tube operating into a plate impedance 
that is small compared with Rp, the effective Ip 
remains essentially constant as the peak-to-peak 
amplitude of E, is increased from a value equal to 
|E„| to a value equal to 2 |Em|, where E„ is the 
cutoff bias. As discussed in paragraph 1-312 in 
connection with the Pierce circuit, the crest ampli- 
tude of Ip between the crest values of E, equal to 

and |Eco| remains approximately equal to 

where Ibm is the zero-bias plate current. There is 
a small maximum approximately equal to 0.54 Im 
(see equation 1 — 812 (21)), but for all practical 


Section I 
Crystal Oscillators 

purposes it can be assumed that Ip is constant for 
all values of gm between class-A and class-B opera- 
tion. Since the effective gm is equal to Ip/E,, the 
doubling of Ep without changing Ip is equivalent 
to halving g,,. If oscillations can be maintained at 
all, the slightest tolerance allowed in gp, ensures 
that the amplitude will build up until the excita- 
tion voltage overlaps the lower bend in the EJp 
curve. Thus, for a sharp-cutoff tube the niinimum 
equilibrium Egp, ( = \/2Eg) will very nearly equal 

Since l/g^ = Zl — R when N = 2, any value 

of Zl greater than 2R„, can ensure that the r-f plate 
current, and hence the output, will be the same for 
all values of R falling within the crystal specifica- 
tions. All that need be done is to design the circuit 
for class-A operation on the assumption that R = 
R,n. This requires the use of a sharp-cutoff vacuum 
tube with tube voltages such that 

Il„ < 2^1^^ 1-412 (7) 

and the design of the load and transformer net- 
work such that 

^ - R„ 1—412 (8) 

Ibm N - 1 

Under the conditions defined by equations (7) and 
(8), a crystal having a maximum R will be driven 
at or under the rated maximum drive, depending 
upon whether Ib„, is equal to, or less than, the value 
specified in equation (7) . If the crystal unit is re- 
placed by another of lower resistance, the crystal 
current and the output will remain essentially the 
same. Since E^ will also be unchanged, the increase 
in the excitation voltage will be entirely that due 
to the decrease in the voltage across the crystal. 
The driving power of the crystal will be directly 
proportional to the crystal R. 

FREQUENCY STABILITY OF 
GROUNDED-GRID OSCILLATOR 

1-413. In the equivalent circuit shown in figure 
1-178 (D), it can be seen that the effective resist- 
ance, Rc, of the crystal circuit is ( 1- R ). But, 

\«m I 

— R* thus 

If N = 2, 

Rc = Zl, = Rl/4 1—413 (2) 


WADC TR 56-156 


269 



Section I 

Crystal Oscillators 

Equations (1) and (2) assume that the mu of the 
tube is large compared with unity, and that 
R|. >> Zi,. From equation 1 — ^241 (2), the frac- 
tional change in frequency required to compensate 
a small change, dtf, in the feed-back phase is 


dw Rc d0 Zi, Ri, do 

T “ 2 VTVC " 2(N - DVXTC “ 8VTV(? 

1-413 (3) 


where L and C are series-arm parameters of the 
crystal, and equation (2) is assumed to hold. Of 
significance is the fact that for a given Zl, the 
frequency stability is independent of the resistance 


of the crystal. However, 



must always be 


greater than R, else the conditions for oscillation 
as defined by equation 1 — 410 (7) cannot hold. 
Thus, although the frequency stability can be con- 
sidered independent of R for a given Zi,, the effec- 
tive Q of the crystal circuit must always be less 


= ^— jof the crystal. 

The larger the g,„ of the tube, the more nearly can 
this limiting value for the effective Q be reached. 


since the more nearly 



can be made to 


approach R,„ in magnitude. But a large g„ must 
be accompanied by a low cutoff voltage for the 
tube, else equations 1 — 417 (7) and (8) cannot be 
made to hold and the crystal will be overdriven. 
Unfortunately — yet not unexpectedly — the re- 
quirements for maximum output are the reverse 
of those for maximum frequency stability. If the 
only frequency-stability problem were to maintain 
the circuit Q as high as possible, the output could 
be increased without decreasing the stability, by 
making both Rl and N large. This could permit 
an increase in Z|. without affecting the value of 

On the other hand, if dm/u in equation 

(3) is to be kept small, not only ^ j^^ -y^but also 

Ae must be kept to a minimum. Because the input 
impedance of the tube is very low, changes in the 
cathode capacitance have a negligible effect on the 
feed-back phase. The principal variations in the 
phase are due to changes in the plate and load 
capacitance. To reduce these effects to a minimum, 
Zi, must be as small as possible. Its smallest per- 


missible value will occur when 


/N-IV 
( N« )* 


IS a maxi- 


mum; that is, when N = 2. Letting N = 2 and 
R,, = 8R„„ Zi, will equal Ri,/4 = 2R„.. The crystal- 
circuit Q for all values of R will then be one-half 


the minimum crystal Q to be expected for the par- 
ticular type of crystal unit. The output will ap- 
proximately equal 2Pp„, for all values of crystal R. 
Much larger outputs can be obtained without 
greatly reducing the frequency stability, by the 
use of remote-cutoff tubes. With these tubes the 
r-f plate current can be made to vary inversely 
with the square root of the crystal resistance. 
Under these conditions, it would be the crystal 
power that remains constant and the output power 
that varies with R. If Pl = 2P<.m when R = R„, 
Pi, = 18P,.„, when R = Rm/9. 

DESIGN PROCEDURE FOR 
GROUNDED-GRID OSCILLATOR 

1-414. The design procedure depends considerably 
upon the special requirements to be met by the 
circuit. As a concrete example assume that a low- 
power, 50-mc oscillator requiring a minimum of 
circuit components and an output amplitude that 
will not be greatly affected by a replacement of the 
crystal unit with another of the same frequency 
is desired. The grounded-grid oscillator is prob- 
ably the best suited for such a purpose. Assume 
further that a frequency tolerance of itO.Ol per 
cent is required without temperature control for 
all temperatures between —40 and -)-90 degrees 
centigrade. Crystal Unit CR-24/U with a fre- 
quency tolerance of ±0.005 per cent between —56 
and -f-90 degrees centigrade should be able to pro- 
vide the required stability. So also will Crystal 
Unit CR-23/U, but the former unit is mounted in 
the coaxial holder, the HC-IO/U, which is gen- 
erally to be preferred because its lower inherent 
shunt capacitance should permit a higher average 
Q. There is no guarantee of this, since the maxi- 
mum Co is 7 ^f in each case; however, the CR- 
24/U employs the 5th harmonic and the CR-23/U 
the 3rd harmonic (thinner crystal) for the 50-mc 
frequency. The greater CR-24/U L/C ratio should 
more than offset its slightly higher R^. Neverthe- 
less, a check should be made to see if crystal units 
of either type having the desired frequency are 
currently being manufactured or have been manu- 
factured in the past. If not, serious consideration 
should be given to the possibility of employing a 
different frequency. The cost of the crystal unit 
will be less if it is already in production, and the 
risk that an undue amount of experimentation will 
be required to produce a crystal unit that meets 
the military standards at an unexplored frequency 
can be avoided. If the crystal unit is expected to 
withstand considerable mechanical shock, the CR- 
24/U must be used, regardless. 

1-416. Assume that a 50-mc CR-24/U crystal unit 


WADC TR 56-156 


270 



has been selected. According to Military Standard 
MS91380, Rm = 75 ohms and P.„, = 2 mw. By 
equation 1 — 412(1) 

(max) Ip = 10®^ ^^^ = 6.2 ma 

According to equation 1 — 412(7) 

Ibn. < 2VX X 5.2 = 14.7 ma 

Assume that the most available tube is the 6AU6, 
sharp-cutoff, miniature pentode. Operated at 250 
plate volts, a Screen voltage of 140 volts provides 
a zero-bias plate current of approximately 15 ma. 
The cutoff bias will be approximately —5 volts. 
Assuming a value of N = 2, by equation 1 — 412(8) 

Zl = 75 -f = 410 ohms 
and 

Rl = 4 Zl = 1640 ohms 
The power ratio when = R^ will be 



The power output for all values of R will be 

Pl = (min) r Pom = 11 mw 

The crystal unit will operate into an effective re- 
sistance equal to 410 ohms. The effective gm of 
the tube will vary from approximately 3000 ^imhos, 
when R is a maximum, to approximately 2500 
fonhos, when R is a minimum. If greater frequency 
stability is required, Ri, can be decreased by ap- 


Sectien I 
Crystal OMillatort 

proximately three-fourths, so that Zl = 300 ohms. 
With this value of Z,„ when R is maximum gm 
will be 4450 /imhos. By increasing the screen volt- 
age to 150 volts or slightly greater, an r-f plate 
current very nearly equal to the maximum permis- 
sible for Rm can be attained. As R is decresised Ip 
and Pl increase somewhat, but the crystal unit 
will not be overdriven. 

MODIFICATIONS OF THE 
GROUNDED-GRID OSCILLATOR 

1-416. Figure 1-179 shows four different designs 
of the grounded-grid oscillator which were built 
and successfully tested at the Georgia Institute of 
Technology. Because of the high initial transcon- 
ductances, 0.011 fimho for the 6J4 and 0.009 for 
the 6AH6, the oscillation amplitude of these cir- 
cuits would drive the average Military Standard 
crystal unit beyond the recommended maximum 
level. This does not mean that a standard crystal 
unit will necessarily be in danger of being shat- 
tered by the circuits shown, but that the frequency, 
resistance, and freedom from spurious modes 
could not be guaranteed by the test standards. To 
employ the circuits illustrated in figure 1-179, dif- 
ferent tubes, or plate-supply voltages may need to 
be used. 

1-417. Figure 1-179 (A) is a narrow-band oscilla- 
tor with the load connected across the secondary 
of the plate transformer. Except for the fact that 
the input “transformer” (Lk, having a 1 : 1 voltage 
ratio) provides no phase reversal, the circuit is 
very similar to that of the basic transformer- 
coupled oscillator. Rl, connected as shown, is 
equivalent to a load resistance of N'^Rl connected 
across Lp. The variable capacitance is for tuning 
out the transformer leakage inductance. Lk is anti- 
resonant with the cathode capacitance, and the 



Figure I- 1 79. Modtflcafions of grounded-grid oscillator. (A) Narrow-band circuit 

271 


WAOC TR 56-156 



Section I 

Cryctol Oecillators 




Figun 1-179. ModMieationa ttf grot$ndmd-gHd escll/otor. (B) Brooft-band eircuH. (C) Circuit for compontatlng 
crystal capacitanco by mutual Inductanco. (D) HIgh-offIcloney class-C circuit 


WADC TR 56-156 


272 



cathode resistance broadens the tuning. The cir- 
cuit was tuned for operation at a center frequency 
of 55 me. The plate transformer consists of 12 
turns of AWG No. 26 PE wire wound on a Miller 


Section I 
Crystal Oscillators 

type 69048 slug-tuned coil form and tapped at 4 
turns. The circuit operates class B. The operating 
data for several different crystal units is given 
below. 


f 

(me) 

Harmonic 


R 

(ohms) 

Stability 
(ppm/ volt) 

n^m 

Pi. 

(mw) 

48 

3 

BH 

45 

0.42 

3 

65 

50 

3 


35 

0.20 

1.5 

103 

54 

3 


45 

0.11 

9 

78 

58 

. 7 


32 

0.14 

0.6 

28 

60 

3 

13 

25 

0.24 

1.6 

40 


The frequency stability is measured in average 
parts per million per volt when the voltage is 
changed by 50 volts. Af gives the deviation ob- 
served between the series-resonance frequency, 
when measured with Cl Meter TS-683/TSM, and 
the actual oscillator frequency. It would seem that 
the 54-mc crystal, which should show the smallest 
value of Af, was influenced by a spurious mode. 
1-418. The circuit in figure 1-179 (B) is designed 
for broad-band untuned operation. Lp consists of 
10 turns of No. 30 PE wire, tapped at 3.3 turns 


and wound on a 0.4-inch-diameter form. The ter- 
tiary winding, Lr, consists of 17 turns on a 0.24- 
inch form that can be slipped inside the L„ core 
by a screw adjustment. The circuit is first tuned 
with a 1000-ohm load connected directly across Lp 
with Lt open. Next, with the circuit connected as 
shown, the coupling between Lt and Lp is adjusted 
until the same grid current as before is obtained. 
All tuning adjustments were made at 57.5 me. The 
performance data of this circuit for several differ- 
ent crystals is given below. 


f 

(me) 

Harmonic 


R 

(ohms) 

Grid Ic 
(/lamp) 

Stability 

(ppm/volt) 


Pl 

(mw) 

48 

3 

14.5 

25 

58 


6 

41 

50 

3 

11 

28 

85 


-0.6 

45 

58.31 

7 

8 

80 

60 


0 

52 

65.31 

7 

10 

80 

65 

0.15 

0.2 

46 

66.65 

5 

4 

65 

90 

0.21 

mSm 

50 

67.2 

7 

14 

80 

62 

0.07 

■1 

47 


1-419. The circuit in figure 1-179 (C) is designed 
to compensate the capacitance of the crystal unit 
by mutual inductance between the plate and cath- 
ode inductors instead of by a shunt inductor as in 
circuit (B). L, consists of 9 close-wound turns of 
AWG No. 30 PE wire, tapped at 2.5 turns; Li, 
consists of 4.5 turns of AWG No. 30 PE wire on 
a thin spacer. The proper coupling adjustment is 
obtained by substituting a capacitance equal to C, 
in place of the crystal and adjusting the circuit to 
oscillate at the true series-resonance frequency of 
the motional arm, but only after L. and L^ have 
separately been adjusted to resonate with Cp and 
Ck, respectively, at their computed resonant fre- 
quencies («>,’ = wb* = 7 ^) • It can be shown 
L.Cp J-<b^k 

WADC TR 56-156 


that 


Lp N,^ Lp 

(1 - M^) (Lp -H N,* L„) 

Lk Na^ Lp 

(1 - M*) (Lk -h N/ Lp) 


1—419 (1) 
1—419 (2) 


and 

Lp U/N.^ 

(Lp -h N,^ Lp) (Lp -H Lk/N**) 

1—419 (3) 


where Lp and Lk are the values of the plate-to- 
ground and cathode-to-ground inductances, respec- 


273 































Section I 

Cryttal Otrillaton 

tively, that would occur if there were no coupling 
between them, L„ is the imaginary shunt induct- 
ance that would be required to antiresonate C„, 
M is the coefficient of coupling between the plate 
and cathode inductors, and N, and Nj are the plate 
and cathode turns ratios, respectively. Nj is simply 
equal to unity in the circuit shown. The perform- 
ance data for the circuit is as follows: 

f = 58.31 me 
stability = 0.28 ppm/volt 
Pl = 90 mw 

Af = operating freq minus tested series- 
resonance freq = —100 cycles 
Af when Ck increased from 10 to 13 /i^f 
= —60 cycles 

Af when C„ increased from 8 to 11 /»/tf 
= —45 cycles 

Af when Cp increased from 6 to 6.5 nid 
= —90 cycles 

1-420. The circuit shown in figure 1-179 (D) is 
designed for high-efficiency operation as a small 
class-C power oscillator. L,, consists of 20 turns of 
AWG No. 28 PE wire wound on a 0.25-inch coil 
form and tapped at 1 turn. Lj, is a 10-turn, 0.25- 
inch-diameter coil of AWG No. 28 PE wire, tapped 
at 5 turns. The observed performance data for this 
circuit is as follows; 

f = 50 me 

load voltage = 9 volts 
grid bias = — 10 volts 
load power =1.9 watts 
crystal power = 0.08 watts 
frequency stability = 0.6 ppm/volt 
plate dissipation = 3 watts 
efficiency = 63 per cent 


concerned. The grounded-plate oscillator can be 
designed for larger outputs by providing a step-up 
transformer in the cathode circuit and removing 
the r-f voltage from the gridleak resistor, as is 
shown in figure 1-180 (B). This permits the cath- 
ode-follower to operate into the same output im- 
pedance but with a greatly reduced load resistance 
across the crystal circuit. The output per milliwatt 
of crystal power is thereby increased. Increasing 
the power output in this manner makes the oscil- 
lator more critical to design and adjust so as to 
prevent free-running oscillations, particularly if 
the tube is to be operated class C, where the effec- 
tive input impedance becomes more or less un- 
predictable at very high frequencies. 

1-422. The over-all gain equation of the oscillator 
in figure 1-180 (A) is 


MNRaZk 

(R Rj) (Rp -)- Zk -f- pZk) 


1-422 (1) 


where Zk is the total effective resistance between 
the cathode and ground. Assuming that the resist- 
ance presented by the transformer is equal to 
Rg/N* and is much greater than R„ we have 


R] (R -i- R 2 ) 

Rj + R + R2 


1—422 (2) 


The effective resistance into which the crystal 
operates is 


Rp = Zk' 4- R -h Rj 1—422 (3) 

where Zk' is the output impedance of the cathode 
follower as faced by the crystal. If is very large 
compared with unity. 


The Grounded-Plote Oscillator 

1-421. The vacuum-tube circuit of the grounded- 
plate oscillator shown in figure 1-180 is essentially 
the same as the two-tube Butler oscillator except 
that a step-up transformer replaces the grounded- 
grid amplifier of the Butler circuit. The gain of 
the Butler grounded-grid tube is thus replaced by 
the gain, N, of the transformer in figure 1-180 (A) . 
The grounded-plate oscillator is most advanta- 
geous when used in the electron-coupled form, as 
shown in figure 1-180 (C), where the plate circuit 
can be tuned to provide frequency multiplication. 
Otherwise, the larger output of the basic trans- 
former-coupled circuit or the greater simplicity of 
the grounded-grid circuit make these oscillators 
preferable to the grounded-plate design insofar as 
obtaining the same order of frequency stability is 


2'‘' = 1 , p 1-422 (4) 

1 + gn. Rl 

For crystal resistances on the order of 75 ohms or 
smaller, R, and R, can also be approximately 75 
ohms each. Values of R, = 68 ohms, R, = 100 
ohms, Rg = 200K, and N = 9 have been recom- 
mended for use with a 6J4 triode. The shunt ca- 
pacitance of the crystal unit, as well as that of the 
grid and cathode, can be compensated if need be 
by conventional antiresonant inductors. To be pre- 
ferred is the method described in the discussion 
of the two-tube Butler circuit — designing the cir- 
cuit so that 

gm Zk = ^ 1--422 (5) 

Uk 


WADC TR 56-156 


274 





Section ! 

Crystal Oscillators 



8 + 

figun 1-181. Traitthron crystal oscillator 


Transitron Crystal Oscillator 

1-423. The transitron oscillator (see figure 1-181) 
operates by virtue of the negative transconduc- 
tance between the suppressor and screen grids of 
a pentode. The total cathode current of the pentode 
is little affected by variations in the suppressor 
voltage, being primarily a function of the poten- 
tial between the screen and cathode. However, as 
the suppressor voltage is made more negative, the 
fraction of the total space current diverted to the 
screen circuit is increased. The screen voltage 
therefore tends to follow the suppressor voltage. 
By connecting a resonant feed-back network be- 
tween the screen and suppressor, oscillations can 
be maintained and no phase reversal is necessary. 
The principal advantage of this circuit is its sim- 
plicity and its ability to oscillate with series-mode 
crystals having comparatively high series resist- 
ances. It can be employed in the v-h-f range, but 
unless the crystal resistances are expected to be 
abnormally high, the relatively large electrode ca- 
pacitances and the small transconductance make 
the performance inferior to that of the trans- 
former-coupled oscillator. 

1-424. When the circuit is used with high-resist- 
ance crystals it is very important that the crystal 
shunt capacitance be properly compensated, in 
order to eliminate the possibility of free-running 
oscillations. As has been demonstrated by W. A. 
Edson with the aid of Nyquist diagrams (graph- 
ical representations of the over-all loop gain and 
phase rotation as the frequency is varied from 0 
to 00 ) , the circuit can be designed to permit only 
one mode of oscillation if, treating g„ as unsigned. 


gL^/ Ci-hC, Cl C,\ 
R. \ R„ ^ R, R, / 


C„ 


1—424 (2) 


The condition implied by equation (1) when gm is 
its maximum possible value means that the loop 
gain is insufficient to start or maintain oscillations 
at any frequency unless R, is effectively decreased 
(such as being bypassed by the series-resonance R 
of the crystal) so that the left side of the equation 
is greater than or equal to unity. Equation (2), 
when satisfied, means that a zero phase shift in 
the feedback can occur at only one frequency. 
Thus, if the circuit is tuned for operation at the 
desired series-mode frequency of the crystal and 
equations (1) and (2) are satisfied, spurious oscil- 
lations will not be possible. 

1-425. Note that Co is effectively increased by the 
suppressor-to-screen capacitance, so that Lo must 
be smaller than would otherwise be the case. C, 
and Cg are simply distributed capacitances to 
ground. Each of the three parallel combinations 
are antiresonant at the crystal frequency, so 

L,C, = L,Co = LgCg = Assuming that the 

antiresonant circuits have impedances Rl, R and 
Rg, respectively, then if Ig, is the r-f screen cur- 
rent, the voltage across the load is 


_ Ig2 Rl (R + Rg) 
Ri, -H R + Rg 

The r-f suppressor voltage is 


1—426 (1) 


Km Hl Rg ^ , 
Ro -f Rl -t- Rg 


1—424 (1) 


_ Eg2 Rg _ 1,2 Rl Rg 

** R -h Rg Rl + R + Rg 


1—426 (2) 


If we assume that E,, is small compared with the 


and 

WADC TR 56-156 


276 





Section I 
Crystal Otciliators 


d-c screen voltage, and define the suppressor-to- 
screen transconductance as the change in screen 
current per change in suppressor-to-cathode volt- 
age — not per change in the suppressor-to-screen 
voltage — the gain conditions for equilibrium are, 
by equation (2), 


Eg3 1 _ Rl 

1*2 Sm Rl + R + Rg 


1-425 (3) 


Of the vacuum tubes available, the 6AS6, which 
has a suppressor-to-screen transconductance of 
1600 fimhos, is probably to be preferred. With this 
tube, oscillations can be maintained with crystal 
units having series resistances of well over 1000 
ohms. Although oscillations can also be main- 
tained with large values of Rl and Rg, these re- 
sistances should be kept as small as practicable so 
as not to unnecessarily degrade the crystal Q and 
reduce the frequency stability. The transitron 
oscillator is also quite useful at low frequencies, 
particularly with high-resistance crystal units. 
When a fundamental-mode crystal element is em- 
ployed, the tuned circuits may not be necessary ; 
but to avoid the possibility of free-running oscil- 
lations or unwanted crystal modes, at least the 
screen circuit should be broadly tuned. (See para- 
graph 1-590 for discussion of negative-resistance 
limiting of transitron circuit.) 


Impedance-lflveriinq Crystal Oscillators 

1-426. Impedance-inverting oscillators employ a 
network similar to that shown in figure 1-182(A) , 
to permit conventional lower-frequency oscillators 
to be operated with crystal control in the v-h-f 
range. A number of these oscillators were de- 
signed and tested at the Georgia Institute of Tech- 
nology under the direction of Mr. W. A, Edson. 
The discussion to follow is based on the final re- 
port of this research. The impedance-inverting 
network is designed to behave as a quarter-wave 
line having a characteristic impedance, Z« = uLi 

= . With this design, the network 

ZJ 

always appears as an inverted Z, equal to Z. = 

where Z. is the series-arm impedance of the crys- 
tal. If Z. = 0, Co is shorted out and Zg is infinite. 
(L, is assumed to have a zero loss.) If Z, is in- 
finite, Lj is series-resonant with Co, and Zo = 0. 
If Z. = Zo, the network appears as an infinite line 
with Zo = Zo. When Z, is a small inductive react- 
ance, Zn is a large capacitive reactance, and vice 
versa. With Zo > > R of the crystal, the network 



Figur* 1-182. Impedance-inverting osciflator circuits. 
(A) Baak impedatKe-inverting network 


serves to invert the crystal resistance to a high 
impedance equal to Zo*/R. For a given C„, maxi- 
mum frequency stability is to be had under the 
quarter- wave line conditions (C„ = C,), but, if 
desired, higher impedances can be had by making 
Co less than C*, or by reducing the effective values 
of Co and C„ with the use of shunt inductors. The 
shunt inductances, however, should be consider- 
ably larger than the values required for antireso- 
nance at the crystal frequency. With «> equal to 
the series-arm resonance frequency, and L, z= 


Z„ appears as an antiresonant re- 

<!> l-'o W 

sistance when the series arm of the crystal is reso- 
nant and Z, = R. At frequencies well removed 
from crystal resonance, the crystal behaves simply 
as a capacitance, C„, .so that the network has a sec- 
ond antiresonant frequency, the square of which 
C 4- C 

is wj’ = -. To ensure that this second fre- 

quency is damped out, a resistance equal to Z, can 
be connected across the crystal unit. 


1-427. Even though the equivalent impedance-in- 
verting network is design^ to be antiresonant at 
approximately the crystal frequency, the operat- 
ing frequency may well require that the crystal 
network facing the actual terminal connections be 
reactive if the necessary phase reversal is to be 
accomplished. For example, it is necessary that 
the actual plate-to-grid network appear inductive 
when used in the Pierce circuit. In the Pierce cir- 
cuit the fundamental modification introduced by 
the impedance-inverting circuit is simply the ad- 


dition of an inductor having a reactance uL, = -7:^ 


in series with the crystal. It can be imagined that 
the reactance of the inductor replaces the X, of a 


WADC TR 56-156 


277 



Section I 

Crystal Oscillators 


parallel-resonant crystal unit* and the low series- 
resonant R of the crystal approximately replaces 
its parallel-resonant value, R,.. C„ of the network 
is C|,„ of the vacuum tube, if we view the circuit 
literally. By this interpretation, C„ is not anti- 
resonant with the inductive branch, but must offer 
a higher impedance than does the inductor at the 
operating frequency. On the other hand, if the 
entire external circuit is viewed in toto by the 
negative-resistance method, which is the imped- 
ance-inverting interpretation, C„ appears as an 
equivalent capacitance equal to 0,,^ plus the addi- 
tional amount required to make the network anti- 
resonant. Since this latter interpretation can be 
employed to illustrate any oscillator circuit that 
contains a crystal connected in series with an in- 
ductance L, = l/<o*C„, the presence of the series 
inductance alone could be sufficient to define an 
impedance-inverting oscillator. The inverted im- 
pedance, Z„, is related to the impedance of the 
inductor and crystal branch as the PI of a crystal 
is related to the equivalent impedance of the crys- 
tal unit. Although these questions are somewhat 
academic, for some readers it may be more helpful 
to interpret the network in figure 1-182 (A) as an 
impedance-converting circuit rather than as an 
inverting circuit. In the transitron circuit, the net- 
work is directly used to invert the crystal R to a 
higher effective resistance, but in other applica- 
tions the designer may prefer to treat the actual 
network as an equivalent X, and R* of a parallel- 
mode crystal unit, transferring the equivalent im- 
pedances directly to the equations of the basic 
parallel-resonant oscillators. 

1-428. There are two significant advantages to the 
impedance-inverting type of design. One is that 
the conventional parallel-resonant circuits can be 
operated with excellent frequency stability in the 
v-h-f range. Another is that by using series-mode 
crystals at the fundamental frequencies, the design 
restrictions regarding the parallel-resonant type 
of crystal unit can be avoided. No data is avail- 
able, but experimentation may show that even in 
the fundamental-frequency range larger outputs 
can be obtained with an inductor and a series- 
mode crystal without degrading the over-all fre- 
quency stability. The chief disadvantage of the 
impedance-inverting network is that it cannot be 
used for broad-band untuned operation. 

IMPEDANCE-INVERTING TRANSITRON 
OSCILLATOR 

1-429. An experimental 50-mc impedance-invert- 
ing transitron oscillator is shown in figure 1-182 
(B). In this circuit, the network, consisting pri- 


marily of L,, Cn, and the crystal, is adjusted to 
present a resistive impedance between the screen 
and ground. Since Ri, is very large compared with 
the crystal R, it can be assumed that Rt, is effec- 
tively connected in parallel with the antiresonant 
network. C„ includes a 3 — 12 /t/if padding capaci- 
tor adjusted at 5 /i/i»f, the screen-to-ground capaci- 
tance, and the suppressor-to-plate capacitance 
(the latter is added because the screen is prac- 
tically bypassed to the suppressor and the plate is 
at r-f ground). E*, in this circuit can be consid- 
ered equal to E,j. Thus, the condition required for 
oscillations to build up is simply that 1/gm be 
smaller than the actual plate-to-ground resistance. 
At the plate voltage used, the initial gm is approxi- 
mately 1500 /imhos, so 1/gm = 667 ohms. Ignoring 
the suppressor-to-ground resistance, the screen 
operates into an impedance of Ri.Z„/(Ri. -f- Z„) = 
1350 ohms, where Z^ = XcnVR. The margin of gain 
is therefore on the order of two to one. The power 
delivered to Rl was observed to be 15 mw. The 
frequency deviation was measured at 0.1 ppm/ 
volt. When Ri was replaced by a 40,000-ohm re- 
sistor, the frequency deviation was found to be 
only 0.004 ppm/volt. Although the power output 
is low, the extraordinary independence of the fre- 
quency under variations in the supply voltage 
marks the impedance-inverting transitron oscil- 
lator as the most stable to use in the v-h-f range. 
One of the chief reasons for this stability is very 
probably the fact that the r-f screen current need 
contain no reactive component. The impedance- 
inverting network, as faced by the screen, can 
appear as a pure resistance. 

IMPEDANCE-INVERTING PIERCE 
OSCILLATOR 

1-430. A 50-mc impedance-inverting Pierce oscil- 
lator is shown in figure 1-182 (C). This circuit 
supplied 70 mw to the 1500-ohm load, and had a 
frequency deviation of 0.6 ppm/volt. Note that the 
total Cb is equal to the total C„. The antiresonant 
Cn for the inductive branch of the impedance-in- 
verting network is thus very nearly (Cp* -j- Cp/2), 
which in turn is equal to Cp. This value of Cn neg- 
lects the equilavent negative capacitance due to 
the reactive component of the r-f plate current. 
Viewed only as an impedance-converting network 
connected between the plate and grid, Cn = Cpg, 
and the network appears as an inductive reactance 
numerically equal to 2/o,Cp or 2/wCg. The upper 
useful limit of this type of circuit is approxi- 
mately 100 me, 

1-431. Figure 1-182(D) shows an electron-coupled 
modification of the impedance-inverting Pierce 


WADC TR 56-156 


278 



Section I 
Crystal Oscillators 


RFC R| 



(B) 

Figure f-182. Impedanee~inverting oeeillator eircuits. (B) Impedance-inverting transitron oscillator. (C) 
Impedance-inverting Pierce oscillater. (D) Impedance-inverting electron-coupled Pierce escillater 


oscillator. The frequency of the oscillator circuit 
is virtually independent of the tuning adjustments 
in the plate circuit. With the plate circuit tuned 
to the 1st, 2nd, 3rd, and 4th harmonics succes- 
sively, the power supplied Rl was found to be 400 
mw at 54 me, 225 mw at 108 me, 50 mw at 162 


me, and 10 mw at 218 me. The frequency stability 
is approximately the same as that of the triode 
circuit in figure 1-182 (C). The upper frequency 
limit of the grounded-screen circuit in (D) was 
found to be 70 me. 


WADC TR 56-156 


279 



Section I 

Cryctal 0*cillatera 



(E) 

Figure 1-187. Impedanee-InvertiHg oatillater cinuita. (i) Impedenn-invrting Miller otclllafer 


IMPEDANCE-INVERTING MILLER 
OSCILLATOR 

1-432. Figure 1-182 (E) is an experimental design 
of a 50-mc, impedance-inverting, Miller oscillator. 
The oscillator is designed so that the C, of the 
equivalent negative resistance circuit is equal to 
Co. Assuming that the reactive component of the 
plate current is negligible, Cn = Co = Cg -f- C„ 
where C, is the equivalent capacitance of Cp, in 
series with the parallel combination of Cp and Lp. 
Cl is thus given by the equation 


Cpg (1 - 0.^ Lp Cp) 

I (Cp -h Cp,) 


1-432 (1) 


Cp, and Cp are fixed by the tube capacitances, and 
C, is equal to C, — C„ so the solution of equation 
(1) requires a definite value of Lp, which in cir- 
cuit (D) was found to be 0.744 ^ih. The circuit 
supplies Rl with a power output of 0.5 watt for 
a crystal drive of 0.07 watt. The frequency devia- 
tion was found to be 0.6 ppm/volt. 


Grounded-Cathode Two-Stage 
Feed-Back Oscillator 

1-433. The two-stage feed-back oscillator (see fig- 
ure 1-183) is used primarily for high-resistance, 
series-mode crystals operating at fundamental 
frequencies not higher than 500 kc and usually 
below 300 kc. The design is rather straightfor- 
ward. V, and V, are tubes of the same type and 
can bb contained in the same envelope. Although 
pentodes should permit slightly greater frequency 
stability, triodes are quite satisfactory for most 
purposes. Since Vj alone can provide the neces- 
sary phase reversal, both tubes can operate into 
resistive loads. The Vj plate circuit is thus tuned 
to the crystal resonance frequency. The proper 
adjustment of Cp is indicated by a maximum read- 


ing on the meter, M. R, is connected across the 
Lp-Cp tank, to broaden the tuning and reduce the 
frequency effects of variations in the Vj plate ca- 
pacitance. The resistance, R„ of the crystal circuit 
is approximately R -f- 2R,. On the assumption 
that R = Rp, (the maximum permissible crystal 
resistance), R, should be made as small as pos- 
sible consistent with stable oscillations. This is 
desirable in order for the effective Q of the crys- 
tal circuit, and hence the frequency stability, to 
be maximum. 

1-434. The loop-gain requirement for equilibrium 
is 


G. G* Ga = . |si = 1 1-434 (1) 

JJigi JJipi JJjpa 

where 

G, = = g„. Ra 1^134 (2) 

L,i 


* Ep, R -I- 2R, 


and 

C = 5si = 

Epa R -I- R, 


1-434 (3) 

1—434 (4) 


Equations (2), (3), and (4) assume that V, and 
Vi operate into plate impedances approximately 

equal to R, and » respectively, and 

that these impedances are very small compared 
with the Rp of the tubes. Combining equations 
(2), (3), and (4), we find that at equilibrium, 
the tube transconductances are such that 


Gi Gj G* 


g»l 8m2 Rl* R 2 
R 2Rj 


1-434 (6) 


WADC TR 56-156 


280 



Section I 
Crystal Oscillators 



figun 1*183. Grounded-cathode twe-ttage feed-back etcillator 


Assuming that ga,, gm* = g„*, where g„ is the nom- 
inal class-A transconductance of the V, and V, 
tsrpe of tube, and that R = Rm, we can select 
values of R, and R^ so that equation (5) will 
equal 1.5. This provides a 3-to-2 margin of gain, 
which should be sufficient to ensure operation with 
all but completely defective tubes. The cathode re- 
sistors can be selected so that the amplitude of 
oscillations does not overdrive a crystal of maxi- 
mum R. An alternative, and possibly a simpler 
approach, is first to select a cathode resistor for 
V„ with the intention of operating that tube at 
a fixed class-A bias. The gain of the Vi stage 
can then be treated as a predetermined constant 
and the V, stage designed to provide the neces- 
sary limiting by gridleak bias. The class-A gain 
of the V, stage must be sufficient to permit oscilla- 
tions when the crystal unit has a maximum re- 
sistance, and the excitation current must not be 
sufficient to overdrive the crystal unit when the 
crystal resistance is a minimum. If desired, a 
parallel-mode crystal unit connected in series with 
its rated load capacitance can be substituted for 
a series mode crystal unit. Such operation in- 
creases the average effective feedback resistance, 
but the presence of the capacitor can reduce the 
tendency of the circuit to oscillate at unwanted 
frequencies. 

MODIFIED TWO-STAGE FEED-BACK 
OSCILLATOR 

1-435. A modification of the two-stage, feed-back 
oscillator to reduce the higher harmonics and 
thereby improve the quality of the sine-wave out- 


put for sync control is shown in figure 1-184. It 
can be seen that the tuned tank, undamped, is con- 
nected in the plate circuit of V, instead of that 
of Vi as is conventionally done. The output, E„ is 
taken from a different part of the tank in each of 
the three circuits represented. The non-bypassed 
cathode resistors are inserted for their degenera- 
tive effect on the higher harmonics and parasitic 
frequencies. They also reduce the effective input 
capacitance of the tubes. It would seem that the 
degradation of the crystal Q is somewhat large. 
The Miller effect in V, is probably significant in 
determining the impedance that the crystal faces 
— certainly so in circuit (C), where R, is one 
megohm and C. is inserted to increase the plate- 
to-grid capacitance by 25 /^if. However, the chief 
purpose of Cg is to serve as a neutralizing ca- 
pacitance for all free-running oscillations where 
the crystal unit would behave as a capacitance, C,. 

Celpitts Oscillators ModHiocI for Crystal Control 

1-436. Figure 1-185 illustrates a number of spe- 
cial-purpose circuits which are basically Colpitts 
oscillators modified for crystal control. Circuits 
(A), (B), and (C) are conventional Cl-meter 
oscillators (see paragraph 1-220) . The tank in- 
ductance, equal to 2L„ is split into two equal in- 
ductances, Lia and L,b. Each of the variable 
inductors in circuit (A) actually represent seven 
fixed inductors which can be connected into the 
circuit by a range switch. The capacitors C, (A 
and B) are continuously variable, and are so 
ganged that Cia is always equal to C,b. Circuits 


WADC TR 56-156 


281 



Saetion I 

Crystal Oscillators 


Fig. 

Equipment 

Purpose 

F, , 

CR 

Hi 

Rr 

Rs 

R« 

Rs 

(A) 

Range Calibrators 
TS-102/AP and 
TS-102A/AP 

Crystal con- 
trol of 500-yd 
marker and 
sync pulses 

327.8 

Sig C Stock 

No. 2X62- 
327.8; WECo 
No. D-168342 

1.8 

10 

18 

1000 

18 

(B) 

Calibrator 

TS-ld/APQ-5 

Crystal con- 
trol of 1000-ft 
marker and 
sync pulses 

491.04 

WECo No. 
D-164868 

1.8 

10 

18 

1000 

1.8 

(C) 

Range Calibrator 
TS»293/CPA-5 

Ose for 
radar IFF. 

P/0 Radar Sets 
AN/CPX-1 and 
AN/CPX-2 

186.3 

Belmont 
Drawing No. 
A-8K-3577 

1.8 

10 

1000 

1000 

1.8 


Circuit Data for Figure 1-184. F in kc. R in kilohms. C in mil. L in Mb. 



U) (Bl (c) 

Figure 1-184. ModiHcotione of two-tlage feed-baek eaci/fotor to improve tine-wave output 


Fig. 

Equipment 

Purpose 

F, 

CR 

IB 

Rs 

Rs 

Rr 

Rs 

Rs 


Rs 

R. 

Rio 

(A) 

Crystal 

Impedance 

Meter 

TS-330/ 

TSM 

Substi- 
tution 
circuit 
for meas- 
uring 
param- 
eters 
of crys- 
tal 
unit 

1-15 

Mili- 

tary 

Stan- 

dard 

quarts 

sr- 

units 

2 2 
each 

22 

1 

0.27 

26 

1 

1 


1 



Circuit Data tor Figure 1-185. F in me. R in kilohma. C in accept where otherwise noted. L in fih. 
WADC TR 56-156 282 































































Saclien I 
Crystal Oscillators 


R. 

Rt 

c. 

C, 

Ci 

C« 

C» 

Cs 

Ct 

C, 

c. 

L, 

mi 

2.2 

2.2 

3800 

100,000 

■ 

390 

10,000 

76 

m 

CO 

■ 

600 

6SN7GT 

2.2 

2.2 

8300 

100,000 

3900 

200 

10,000 

75 

10 

3900 

■ 

380 

6SN7GT 

1 

CO 

6000^ 

60,000 

SSO 

76 

■ 

1 

26 

00 

25 

SigC 

Stock No. 

2C-638- 

ICl 

6SN7GT 



ffguM I-I85. ColpMts circuits modMor/ tor sortos-modo crystal control 


Cl 

C] 

c. 

c« 

C, 

Ct 

Ct 

Cl 

C, 

Li 

1 L: 1 

La 

a 

V| 


7 . 8 - 

61 

1 . 6 - 

260,000 

260,000 

10,000 

80 

6-42 

250,000 

2 . 6 - 

H|| 


|■||||| 

6 V 6 GT 

■■i 

140 


7 



386 






Mch 









each 

1 

1 

1 


1 


WADC TK 56-1 M 






































































0^ 0m 

&600 


Equipment 

Purpose 

Crystal 
Impedance 
Meter TS« 
683/TSM 

Substi' 
tution 
circuit 
for meas' 
uring 
param- 
eters of 
crystal 
unit 

Crystal 
Impedance 
Meter TS- 
683/TSM 

Substi- 
tution 
circuit 
for meas- 
uring 
param- 
eters 
of crys- 
tal 
unit 

Test Set 
TS-250/ 

APN 

Range 

osc. 

Output 
consists 
of pos- 
itive 
range 
pips 

Diversity 

Receiving 

Equipment 

AN/FRR-3 

BFO 

with 

sfc; 

manual 

or 

crystal 

opera- 

tion 


CR 

R. 

Mili- 

tary 

Stan- 

dard 

quartz 

crystal 

units 

0.1 

each 

Mili- 

tary 

Stan- 

dard 

quartz 

crystal 

units 

0.0 

56 

each 

Bliley 

No. 

122- 

5006 

(octal 

base) 

100 1 

1 


R. R. I R. I R, I R. I R. 


15 25 


15 25 2.2 


10 20 


5 0.035 500 1.6 


Circuit Data for Fig^ure 1-185. F in kc. R in kilohms. C in except where otherwise noted. L in <ih. 

WADC TR 56-156 284 




























































WADC TR 56-156 


285 



































Section I 

Crystal Osdilators 

(B) and (C) are substantially of the same basic 
design as circuit (A). Except for resistors R« and 
capacitances C3 and C4, the parameters of circuits 
(B) and (C) have the same numbers as their 
functional analogues in circuit (A). Circuits (B) 
and (C) are not designed for parallel-resonance 
measurements. For crystal resistance measure- 
ments, the calibrating resistor must be substituted 
externally for the crystal unit. The capacitors Ci 
are fixed and the inductors L, are continuously 
variable and are so ganged as always to be equal. 
In each of the Cl-meter circuits shown, it can be 
seen that if the resistance of the tank, including 
the crystal, were zero, and if the tank were per- 
fectly balanced, no voltage would exist between 
the crystal and ground. The voltage across Lia 
plus that across C,a would equal zero, and no cur- 
rent would flow through the resistors Ri, which 
effectively form a bridge between the inductance 
arm to the grounded connection of the capacitance 
arm. In practice, a net voltage does exist across 
L,a and CiA in series, and this voltage appears 
across Ria, being measurable at the jacks J1 and 
J2 in circuit (A). The r-f voltage across Rib is 
approximately that across J3 and J4, which in 
turn is equal to the Ria voltage plus that across 
the crystal resistance. The Ri resistors are not 
essential insofar as maintaining oscillations is con- 
cerned, but they load the circuit, thereby reducing 
the effect of the variations in crystal resistance 
upon the oscillator activity, and they serve to pro- 
tect the crystal, to balance the circuit to ground, 
and to facilitate measurements of the crystal volt- 
age (Eiie = Ej 3 — Eji) without unduly interfering 
with the effective circuit parameters. The Cl-meter 
oscillator can be analyzed as a particular type of 
transformer-coupled oscillator, as an impedance- 
inverting oscillator, or as an equivalent Pierce 
oscillator having a crystal X,. =: a>(L,A + L,n) and 
an effective crystal resistance accounting for the 
losses in the resistances Ri as well as in the R, of 
the actual crystal. 

1-437. Figures 1-185 (D) and (E) are examples 
of grounded-plate Colpitts circuits which have 
been modified for series-mode crystal control. Cir- 
cuit (D) is designed to provide positive range 
pips to the grid of Vj. The circuit operates class C 
at either one of two frequencies, the appropriate 
crystal being connected between the cathode tank 
and the grid of V,. Circuit (E) is designed for 
either manual or crystal control. During manual 
control the resistor R, replaces the crystal unit. 
V, is operated as a reactance tube. The a-f-c bias 
varies in such a way that the b-f-o frequency 
tends to follow any changes in the frequency of 


the teletype signal being received. 

CRYSTAL CALIBRATION 

1-438. The design of a crystal oscillator to be 
used for calibrating the frequency of other oscil- 
lators generally is directed toward obtaining out- 
puts rich in harmonics. Where tuned-plate circuits 
are required the L/C ratios should be high, so 
that high impedances are also presented to the 
overtone frequencies. The oscillator should be 
operated class C, and often the gridleak resistance 
is a megohm, or higher. If the crystal calibrator 
is to serve as a frequency standard of greater- 
than-average precision, this precision becomes the 
principal design problem insofar as the oscillator 
is concerned; if need be, the required harmonics 
can be developed in nonlinear amplifier stages that 
follow the oscillator stage. The higher the over- 
tone, the weaker will be its effective output power, 
but with proper design useful outputs up to and 
above the 100th harmonic can be obtained. With 
the addition of frequency multiplier and/or di- 
vider circuits a single crystal can provide a useful 
calibrator frequency range as broad as desired. 
For maximum precision, a G element, usually cut 
for 100 kc, should be used. 

1-439. Figure 1-186 illustrates a simply designed 
crystal calibrator employing an electron-coupled 
Miller oscillator operating into a resistive plate 
load. Such a circuit will ensure sufficient frequency 
stability for most purposes. Harmonic outputs in 
steps of 100 kc are provided up to frequencies of 
10,000 kc. For higher frequencies, the 1000-kc 
crystal can be used to provide calibration points 
in multiples of 1000 kc. The variable grid capaci- 
tor is employed to ensure that the crystal operates 
into the correct load capacitance. 

Crystal Calibrator Employing Regenerative 
Freguency Divider 

1-440. Figure 1-187 shows the regenerative fre- 
quency-divider circuit of the crystal frequency 
indicator (CFI) used in Radio Transmitting Set 
AN/ART-13A. This circuit employs a 200-kc crys- 
tal to control a rich mixture of harmonics, pro- 
viding useful check points spaced as close as 25 kc 
apart. The crystal oscillator, utilizing the triode 
section, V„ seems best described as a modification 
of an impedance-inverting Pierce circuit. When 
oscillations first start, the output of the oscillator 
is fed to grid No. 1 of the pentagrid mixer, V3. 
The 50-kc and 150-kc components of the noise volt- 
ages that are mixed with the 200-kc signal are 
amplified by V3 and fed to the input of the Vj 
triode section. The V, plate circuit, which is tuned 


WADC TR 56-156 


286 



Section I 
Crystal Oscillators 


I lOOKe 5 iOOO S I'SMEO S I00> 
X Kc > >2000 

> <0HM8 




* eooo- • **9® 


Ngun I- 166. Typico/ des^n of sinplo-tvho gonorol-purposo crystal calibrator circuit 


Fa«Fl 'F2-S0KC 


aOOKCag^B^ 


Rl f If 

•OOK 30 


T C7 

04 Mf 


Whsskc 


N0TE = 

UNLESS otherwise SHOWN 

ALL CAPACITORS ARE IN MtCROMiCROFARAOS 
ALL RESISTORS ARE IN OHMS. 


SPEECH AMPL 
INPUT 



figure I-I87. CFf regefierative froquoncy divider In Radio Transmlttor T-47A/AftT*l3 (P/O Radio 

Transmitting Sot AN/ART-13AJ 


WADC TR 56-156 



Section I 

Crystal Oscillators 


to 150 kc, amplifies the 150-kc noise input and 
triples the 50-kc input. The 150-kc output of Va is 
then fed back to the pentagrid mixer at grid No. 3. 
It is again amplified and fed back to Va. How- 
ever, the direct amplification and regeneration of 
the 150-kc signal alone is not sufficient nor prop- 
erly phased to maintain oscillations at this fre~ 
quency. The 150-kc oscillations are sustained prin- 
cipally by tripling the 50-kc feedback, which builds 
up as the amplified difference frequency of the 
200-kc and 150-kc inputs to Vs. The output of Vs 
is effectively a 50-kc fundamental frequency stand- 
ard of large harmonic content that is fed to the 
grid of triode section V 4 , where it is mixed with 
signals from the variable oscillators of the trans- 
mitter. The output of V 4 is fed to the input of an 
audio amplifier, which amplifies the beat note 
whenever the variable oscillator approaches the 
frequency of one of the CFI harmonics. In prac- 
tice, the recommended check-point harmonics are 
spaced 25 kc apart in the 200— ^00-kc frequency 
range, 100 kc apart from 2000 to 3000 kc, 150 kc 
(3000—4000 kc), 200 kc (4000—6000 kc), 300 kc 
(6000—9000 kc), 450 kc (9000—12,000 kc), and 
600 kc (12,000 to 18,100 kc). The presence of the 
harmonics of an apparent 25-kc fundamental, 
which is used in the low-frequency calibrations, is 
not readily explained on the basis of the foregoing 
discussion of the circuit. A complete analysis of 
the nonlinear characteristics of the circuit is not 
available, but it appears possible that if a 25-kc 
signal appears at the plate of Va, it can conceiv- 
ably be sustained by being fed to V^, mixed vnth 
f. to form a sum frequency of 175 kc, fed back to 
Vn and mixed with the 200 -kc injector signal to 
regenerate a difference frequency of 25 kc. It 

3 

should be understood that f^ — 3f.s =-i-fi is a nec- 

4 

essary relation, and that f^ and fa are synchro- 
nized and controlled by the crystal oscillator. The 
phase and frequencies of the regenerative circuits 
automatically follow the phase and frequency of 
the Vi output. For a more analytical study of re- 
generative frequency dividers, see discussions by 
R. L. Miller, R. L. Fortescue, and W. A. Edson. 

SYNTHESIZING CIRCUITS 

1-441. Of great promise, particularly for use in 
airborne radio equipment in the v-h-f range where 
crystal control is necessary to maintain the re- 
quired frequency stability, has been the develop- 
ment of synthesizing circuits, in which a very few 
crystals are able to control a large number of 
channels. In the discussions to follow we shall use 
the term frequency synthesis very loosely to apply 


to any type of frequency-control circuit or system 
in which a few fixed-frequency oscillators are 
used to control or to stabilize a large number of 
radio frequencies. If the term were used rigor- 
ously, it would apply only to those cases where an 
output frequency is produced entirely from heter- 
odyned combinations of internally generated fre- 
quencies. Examples of this type are provided by 
the Plessey frequency generator and by the Collins 
transmitter frequency-control system employed in 
Radio Set AN/ARC-27. For our purposes we shall 
extend the term to cover such systems as the 
Bendix frequency-control circuit in Radio Set 
AN/ ARC-33, where the output frequency is not 
actually synthesized but is obtained from a vari- 
able-tuned master oscillator that is crystal-stabi- 
lized at many frequencies. Also implied by the 
term will be such systems as the Collins crystal- 
controlled multichannel receiver circuits. In these 
latter circuits only one end-product frequency is 
desired — a fixed superheterodyne intermediate 
frequency. But the system design is such that with 
the use of a very few crystals the desired inter- 
mediate frequency can be synthesized under crys- 
tal control from received signals on any one of 
hundreds of possible radio channels, t 

The Plessey Synthesisiiig System* 

1-442. The first crystal-controlled frequency syn- 
thesizer in commercial usage appears to have been 

f Not all types of synthesizers in current use are covered 
in the above discussion. Other recently developed and 
equally important circuits include: 

The General Radio Company synthesizer, developed 
under SiCTal Corps Contract No. DA-86-039-SC-16642. The 
GR synthesizer operates on a principle fundamentally 
different from those described in this report. In the GR 
system an oscillatoi' is ^ase- and frequency-locked 
through a variable scale-of-N divider. The pulse output 
of the divider is compared by coincidence methods with 
a pulse derived from a crystal oscillator. The frequency 
range of this synthesizer is 0.1 me to 10 me. 

The Matawan Synthesizer ME-447, of the Lavoie 
Laboratories Insti'ument Company. This system generates 
any multiple of 1 kc within the range of 1.0 me to 2.0 me. 

The Rohde and Schwarz decade synthesizer and exciter 
system (Federal Telephone and Radio Company Types 
HS-431, HS-441, and HS-471), which covers a range of 
50 kc to 30 me. 

The Telefunken Precision Frequency Meter. This meter 
is used in the measurement of frequencies between 1 kc 
and 300 me. The circuitry contains a frequency synthesizer 
capable of generating sine-wave outputs between 1 kc and 
30 me. It is claimed that harmonics and sidebands of the 
output frequency are at least 80 db below the selected 
signal, and that the syntheslizer accuracy is ± 0.2 cps for 
frequencies between 1 kc and 3 me, and ± 2.0 cps between 
3 me and 30 me. 

* Note: The discussion of the Plessey synthesizing system 
is based primarily upon the report, “The Frequency 
Synthesizer”, by Mr. H. J. Finden of the Plessey Com- 
pany, Ltd., England, published in the Journal of the 
Inutitution of Electrical Engineers, Vol. 90, Part III, 1943. 


WADC TR 56-156 


288 



that developed by the Plessey Company, Ltd. of 
England. This synthesizer has been designed as a 
frequency generator to be used in making precise 
radio-frequency measurements. The synthesizing 
system employed is nevertheless quite applicable 
for other uses, such as providing multichannel ex- 
citation voltages for radio communication equip- 
ment. As designed by the Plesaey engineers, the 
synthesizer generates a sequence of harmonic sig- 
nals of much greater precision and purity than is 
obtainable with conventional tjrpes of frequency 
generators. The original model permits a direct- 
reading dial selection of any of the first 10,000 
harmonics of 1 kilocycle per second; a later and 
larger model extends the range to the first 100,000 
harmonics, i.e., any harmonic of 1 kc up to 100 
me. Ail these frequencies are made available singly 
as pure sine waves (unmixed with other har- 
monics or frequency products) by a decade system 
cf frequency dividers and multipliers, mixing 
stages, and filters where all the generated fre- 
quencies are under the control of a single 1000-kc 
precision crystal standard. Theoretically the sys- 
tem could be extended to cover a broader or a dif- 
ferent frequency range; or could be changed to 
permit steps between adjacent frequencies that 
are smaller or larger than 1 kc. If required, it 
would be quite practicable for the Plessey gener- 
ator, itself, to be expanded to cover also the 100- 
to-lOOO-mc range in 10-mc steps. In 1965 the 
Schomandt Company of Munich, Germany placed 
on the market a similar type of synthesizer fre- 
quency generator covering ^e 0 — 30-mc range in 
1-kc steps. The output of the Schomandt synthe- 
sizer is equivalent to that of the Plessey synthe- 
sizer in quality, having at least a 60-db attenua- 
tion of all unwanted frequencies. It was the 
demand for such narrowly spaced pure output fre- 
quencies for use in making frequency measure- 
ments that originally led to the development of 
the synthesizer circuits. 

SYNTHESIZER ADVANTAGES IN RADIO- 
FREQUENCY MEASUREMENTS 
1-443. Prior to the development of the frequency 
synthesizer there were two conventional methods 
for measuring radio frequencies — the “interpola- 
tion” method and the “successive heterodyning" 
method.* Briefly, the interpolation method, which 
is satisfactory where extreme accuracy is not re- 
quired, consists of mixing the unknown frequency 
with the two nearest harmonics of a frequency 
standard, and zero-beating the difference frequen- 
cies obtained against the output of a linearly 
tuned variable oscillator. It is then possible to 
interpolate the unknown frequency by determin- 


Section I 
Crystal 'Oscillalort 

ing its relative position between the known har- 
monics of the standard. In the successive hetero- 
dyning method, the unknown frequency is mixed 
with a known harmonic of a frequency standard ; 
the difference frequency is then heterodyned with 
a second standard harmonic to obtain a second 
and lower difference frequency; and the process 
is repeated, if necessary, until a difference fre- 
quency is obtained that lies within an accurately 
measureable audio range. Although the successive 
heterodyning method can be quite accurate, occa- 
sions arise where the operator cannot be certain 
without undue checking that the difference fre- 
quencies being measured are not the products of 
unwanted harmonics contained in the heterodyned 
signals. The use of a frequency synthesizer that 
permits individual pure sine-wave outputs of a 
sequence of narrowly spaced frequencies, instead 
of a simultaneous mixture of many harmonics, 
can be said to offer a third and greatly superior 
means of measuring radio frequencies. 

1-444. With the use of decade dial control, greater 
operating simplicity is possible than with the in- 
terpolation method ; and when the pure sine-wave 
frequencies are spaced only 1 kc apart, the inter- 
polation accuracy of the successive heterodyning 
method is maintained, but with the elimination of 
those chance difference products that can result 
from harmonic mixtures of multiple stages of 
heterodyning. 

FUNCTIONAL OPERATION OF PLESSEY 
SYNTHESIZER 

1-445. The circuit system by which the Plessey 
synthesizer produces thousands of frequencies, all 
controlled by a single 1000-kc crystal standard, is 
illustrated in figure 1-188. The block diagram 
shown is that of the original, single-cabinet model 
that permits the operator a choice of any one of 
the first 10,000 harmonics of 1 kc. It can be seen 
that there are three successive stages in which the 
input frequency is divided by 10, so that the last 
divider represents an over-all division of the orig- 
inal standard (1000 kc) by 1000. The dividers and 
the 1000-kc harmonic generator are of the syn- 
chronized, free-running, multivibrator type whose 
outputs are rich in harmonics. Each of these mul- 
tivibrator circuits forms the first stage of a 
sequence which can be tuned to pass any one of 
the first 10 harmonics of its respective multivi- 
brator fundamental. These sequences are labeled 
A, B, C, and D in figure 1-188. In the synthesis of 
a frequency, we can say generally that sequence A 

* Note: See paragraph 2-66 to 2-151 for detailed descrip- 
tions of frequency'-nreasuring systems in current use. 


WADC TR 56-156 


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SYNTH E S I Z ING C I R C U I T S 


Section I 

Crystol Oscillators 


A 

CIRCUITS 


B 

CIRCUITS 


C 

CIRCUITS 


D 

CIRCUITS 



Figuro 1-188. Blotk diagram of a P/ossoy tyntho$iior dotignod to cover the 0 — 10-mc spottrum In 1-kc steps 


supplies that part of the final frequency which is 
a multiple of 1000 kc, B that part which is a mul- 
tiple of 100 kc, C that part which is a multiple of 
10 kc, and D that part which is a multiple of 1 kc. 
1-446. For example, assume that an output fre- 
quency of 6789 kc is desired. The A, B, C, and D 
harmonic selectors, respectively, will be decade- 
set to pass the 6th, 7th, 8th, and 9th harmonics 
of their respective input signals from the preced- 
ing multivibrator stages. In balanced modulator 
C, the output of selector D, 9 kc, is mixed with 
the 80-kc output of selector C. (The signals are 
heterodyned in a balanced modulator circuit 
rather than in « more efficient type of mixer in 
order to eliminate the two input frequencies from 
the modulator output. In this manner the sum and 
difference products become the dominant frequen- 
cies in the modulator output.) Filter C is dial-set 
to pass the desired frequency product, 89 kc, 
which it feeds to balanced modulator B. In modu- 
lator B, the 89-kc signal is heterodyned with the 
700-kc output of the decade-set harmonic selector 
B. Filter B is dial-set to pass the sum product, 789 
kc, from the B modulator output to the A modu- 
lator input, where it is mixed with the 600()-kc 
output of harmonic selector A. Filter A is dial-set 
to pass the sum product, 6789 kc, of the mixed 
signals, which product is then amplified and fed 


through a phase inverter to the synthesizer out- 
put jack. 

1-447. The foregoing example of the operation of 
the Plessey sjmthesizer suggests that the sum 
rather than the difference products of the mixed 
signals are always selected. In practice this is not 
the case, even though the decade dialing system is 
so designed that the operator is always provided 
a direct reading of the output frequency as if he 
were only adding the decade units together. In 
order to sufficiently filter out the unwanted prod- 
uct, it is important that the signals to be mixed 
are so selected that there is at least a 10 per cent 
difference in frequency between the sum and dif- 
ference products. Since the filters must be capable 
of suppressing ail adjacent harmonics of the mixed 
signals, it can be assumed that they are also cap- 
able of suppressing the unwanted heterodyne 
product if it differs from the desired product by 
as much as the fundamental harmonic of the mpd- 
ulator input from the harmonic selector. For ex- 
ample, in modulator C, the space between the sum 
(fc -b fi>) and the difference (fc — fu) frequen- 
cies should not be less than 10 kc, the fundamental 
of the harmonic from selector C. Since 

(fc -f- f,.) — (fc — fi.) = 2f„ > 10 kc 
then fr, must never be less than 6 kc if it is to be 


WADC TR 56-156 


290 



















Section I 
Crystal Oscillators 


mixed with fc. Similarly, fd, must not be less than 
50 kc if it is to be mixed with fs, and fnci> must 
not be less than 500 kc if it is to be mixed with fA. 
1-448. To illustrate, let us suppose that a fre- 
quency of 91 kc is desired. It would not do for fc 
and fi) to be 90 kc and 1 kc, respectively, for then 
the sum product, 91 kc, would be separated from 
the difference product, 89 kc, by only 2 kc. Rather, 
100 kc should be selected as fc and 9 kc as fa. The 
variable filter C would be set to pass the difference 
product, 91 kc; which product differs from the 
sum product, 109 kc, by 18 kc, well beyond the 
minimum permissible limit of 10 kc. 

1-449. As a more involved example we shall de- 
termine the heterodyne frequencies that would be 
used in the synthesis of an 8136-kc output. For a 
mental calculation of the correct frequency com- 
binations the easiest method is to start with the 
output frequency, fABcu, and from this determine 
fA, fnei), fa, fcii, fc, and f„ in that order, working 
from the larger units to the smaller. Each of the 
above six frequencies is determined by remember- 
ing that none of the input frequencies to the A, B, 
and C modulators can be less than 500, 50, and 5 
kc, respectively. Thus, we see at once that 8136 kc 
is not to be the sum product of 8000 kc and 136 kc 
in the A modulator, since 136 kc is less than 500 
kc. So f^ must be 9000 kc and f„ci) must be 1000 
minus 136 kc, that is, 864 kc; which means that 
filter A will be adjusted to pass the difference 
product (9000 kc minus 864 kc). Since 64 kc is 
greater than 50 kc, the required 864-kc output of 
modulator B can be obtained as the sum product 
of 800 kc and 64 kc, fn and fen, respectively. Since 
4 kc is less than 5 kc, the required 64-kc output of 
modulator C must be obtained as the difference 
product of 70 kc and 6 kc, fc and f,,, respectively. 
We see that in order to select an output of 8136 
kc, the decade dials of the A, B, C, and D har- 
monic selectors must be set to pass, respectively, 
the 9th, 8th, 7th, and 6th harmonics. In other 
words, the output frequency would be a synthetic 
product of the four frequencies, 9000 kc, 800 kc, 
70 kc, and 6 kc. So also would be an output fre- 
quency of 9876 kc. Since the decade dials that con- 
trol the harmonic selectors may be set at the same 
positions for two or more frequencies, some ar- 
rangement must be made so that the decade read- 
ing presented to the operator identifies correctly 
the particular frequency being synthesized. This 
convenience is accomplished in the Plessey syn- 
thesizer by manually operated range adjustments 
that alter the correspondence of the dial readings 
with the dial positions. Thus, in the example 
above, with the proper range settings, decade dial 


A in position 9 would give a reading of 8, decade 
dial B in position 8 would give a reading of 1, 
decade dial C in position 7 would give a reading 
of 3, and decade dial D in position 6 would give 
a reading of 6. The mechanics of exactly how 
this feature is incorporated in the Plessey syn- 
thesizer, although relatively simple in principle, 
is somewhat beyond the subject matter of our 
assignment here. 

CIRCUIT DESIGN OF PLESSEY 
SYNTHESIZER 

1-450. The general circuit design employed in a 
Plessey synthesizer is shown in the schematic dia- 
gram of figure 1-190. The circuit shown, when 
synchronized by a 1000-kc standard (whose cir- 
cuit is not shown), is capable of covering the 0 — 
10-mc range in 1-kc steps. Note that each of the 
four decade harmonic sequences begins with a 
multivibrator-type of harmonic generator. Rheo- 
stats are furnished for adjusting the natural oscil- 
lation period of each multivibrator, in order to 
allow for aging effects and the like. More elab- 
orate or reliable harmonic-generator circuits are 
not required since the failure of any of the multi- 
vibrators would be immediately apparent by the 
reading in the output meter. Figure 1-189 shows 
in detail the circuit parameters of the 100-kc mul- 
tivibrator, which also acts as the 1st divider. Note 
that the 1st divider is synchronized by the output 
of the 1000-kc amplifier and not directly by the 
frequency standard. The output of the 1st divider 
in turn is used to synchronize the 2nd divider, and 
that of the 2nd to synchronize the 3rd. 

1-451. In figure 1-190 it can be seen that harmonic 
selection is achieved by switching to the correct 
tuning capacitor from a bank of 10. The same 
inductance is used for each of the harmonics. 
Since the percentage difference between adjacent 
harmonics is less as the order of the harmonic be- 
comes higher, it is more difficult to eliminate the 
9th and 11th harmonics when selecting the 10th,. 
than it is to eliminate the 1st and 3rd when select- 
ing the 2nd. For this reason, the value of each of 
the fixed tuning inductors is chosen to provide an 
optimum Q at the 10th harmonic. This permits a 
relative magnification of the 10th harmonic over 
its adjacent harmonics of approximately 200, 
which is equivalent to a 32-db attenuation of the 
9th and 11th harmonics and more than that for all 
others. The attenuation of adjacent harmonics be- 
comes greater as the selected harmonic becomes 
lower, so that in any event it is never less than 
32 db at each tuned circuit. Two tuned circuits in 
series provide more than a 60-db attenuation. 


WADC TR 56-156 


391 



Section I 

Crystal Oscillators 



Figure 1-189. Schematic diagram of a Plessey syn- 
thesizer designed to cover the 0 — lO-mc spectrum in 
I -he steps. The circuit of the crystal oscillator standard 
is not shown 


which for all practical purposes is sufficient to 
consider the selected harmonic a pure sine wave. 
1-452. With the use of a balanced modulator it is 
not necessary to use as many tuned circuits as 
would otherwise be necessary to eliminate all un- 
wanted harmonics and frequency products. Note 
in figure 1-190 that the balanced modulator design 
is such that two matched amplifiers have a com- 
mon output circuit, but that they are excited by 
equal signals 180 degrees out of phase, so that the 
amplified signals cancel each other in the load. 
Thus, even though the heterodyne efficiency of the 
balanced modulator is less than that of other types 
of mixers, the balanced circuit is greatly advan- 
tageous in helping to eliminate all the unwanted 
frequencies, particularly the unwanted harmonics, 
that originate in the circuits preceding a mixer 
stag!'. In the Plessey synthesizer it can be seen 
that the modulators are provided with a switching 
arrangement by which one of the tubes of each 
modulator can be cut out of the circuit by opening 
its cathode return. One of these switches is opened 
whene\ or a modulator stage must pass an un- 
mixed signal. With one tube removed, the balanced 
arrangement is destroyed, and since only one in- 
put signal is being handled, the vacuum tube still 


connected in the circuit will be operated as a con- 
ventional amplifier. If, for example, the desired 
output were a 2000-kc signal, none of the modu- 
lators would be in operation except modulator A, 
which would be unbalanced and operated simply 
as an amplifier of the 2nd harmonic from the 
1000-kc harmonic generator. 

1-453. Variable-tuned circuits are provided as 
bandpass filters. These must be adjusted manually 
in selecting the proper heterod 3 me product to be 
passed. The selectivity is sufficient to provide at 
least a 30-db attenuation of any unwanted signal 
that differs as much as 10 per cent from the de- 
sired signal. 

1-454. The phase inverters are inserted for proper 
impedance matching. They permit an output at 
any frequency within the operating range of 100 
millivolts across a 75-ohm load. The system as a 
whole insures at least a 60-db attenuation of all 
unwanted frequencies. 

The Beadix Synthesisiiig System 

1-466. In America, much of the pioneering in the 
field of frequency synthesis has been done by the 
research staff of the Bendix Corporation. 

The following discussion is based upon the syn- 
thesizing circuit originally described by W. R. 
Hedeman of Bendix in the magazine Electronics. 
Figure 1-191 shows a block diagram of the syn- 
thesizer circuit developed At Bendix for use in 
controlling the frequency of a continuously vari- 
able v-h-f receiver heterodyne oscillator. In this 
circuit, the first crystal oscillator employs but one 
crystal. The harmonic generator that follows this 
oscillator produces a rich output of harmonics, the 
first of which is f^, the fundamental of the first 
crystal oscillator. The harmonics selector is com- 
posed of a number of band-pass circuits, each cir- 
cuit designed to pass a particular harmonic of the 
crystal frequency. The number of frequencies con- 
trolled by the synthesizer is directly proportional 
to the number of harmonic channels in the selec- 
tor. Let fh equal the harmonic selected and f„ equal 
the frequency of the variable oscillator. The value 
of f„ is always higher than that of fh. These two 
frequencies are mixed in the first frequency con- 
verter to form the sum-and-difference frequencies, 
which, in turn, are fed to the input of the first 
band-pass amplifier. The first band-pass amplifier 
amplifies and passes only the difference frequency, 
f„ — fi,. This difference frequency is fed to the 
second frequency converter, where it is mixed with 
the output, f„ of the second crystal oscillator. The 
second crystal oscillator is generally provided with 
more than one crystal unit, but only the funda- 
mental frequency of the oscillator is used when a 


WADC TR 56-156 


292 




WADC TR 56-156 



Section 
Crystal Oscillatoi 




D CtRCUITS 


Figure 1-190. Schematic diagram at the 100-kc synehroniied multivibrator used in thu Piessey.synfhesizer at a 
^ decade divider of a 1000-kc $tandard and at a lOO-kc harmonic generator 

n 293-294 




SccHon I 
Crystal Oscilloters 



Figure 1-191. Block diagram of froquoney-tynthotixor circuit 


particular crystal is selected. The number of con- 
trolled channels is directly proportional to the 
number of second-oscillator crystals. The sum and 
difference frequencies of the frequency converter 
are fed to the second band-pass amplifier, which 
amplifies and passes only the difference frequency, 
<f„ — fh) — f.. This difference frequency is fed 
to a discriminator. The number of channels con- 
trolled is directly proportional to the number of 
discriminators used. The d-c a-f-c output of the 
discriminator is used to control the bias of the 
control tube. The plate current of the control tube 
determines the rotor position of a servo motor, 
and the rotor is mechanically coupled to control 
the tuning: elements of the variable oscillator. The 
servo motor continues to turn and thereby con- 
tinues to change the frequency, f„, until the output 
of the discriininator is zero. This occurs when the 
output of the second band-pass amplifier is equal 
to frt, the frequency of the discriminator circuit. 
By reversing the polarity of the discriminator out- 
put leads, f„ can be made to vary in the opposite 
direction in order to reach equilibrium. Thus, for 
each value of fh, f„ and fd, there are two equili- 
brium values of f„. These are given by the equation 

f„ = fh + f. ± fd 1^66 (1) 


1-466. Let 

N = Total number of channels, f„. 

H = Number of harmonics used (1st crystal 
oscillator) . 

X = Number of crystals (2nd crystal oscillator). 
D = Number of discriminators. 


It can be seen from equation 1 — 456 (1) that 

N = 2HXD 1—456 (1) 

The factor 2 is introduced by the fact that for each 
discriminator there are two values of f„ for each 
combination of fh and f,. One value is f„ = fh + 
f, + fd, and the other is f„ = fh + f, — fd. 

1-457. As a concrete example, let us imagine that 
it is desired to cover the frequency range between 
100 and 156 me with the channels spaced 200 kc 
apart. The lowest value of f„ is to be 100.2 me, 
and the highest is to be 166 me. Thus, 

N = — (min) f« ^ ^ _ 156 — 100 _ 

Afo 0.2 

1—467 (1) 


By equation 1 — 456 (1), 

HXD = = 140 1—467 (2) 

The smallest total number of elements occurs when 
H, X, and D can be made as nearly equal to each 
other as possible, but in an actual design problem, 
this may not be the most practical solution. In our 
particular example let us assume that the first 
seven harmonics of L are to be used. 

With H = 7, then by equation (2) 


XD = .3^ = 20 


1-457 (8) 


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section I 

Crystal Oscillators 


The combinations (X,D) possible are (20,1), 
(10,2), (5,4), (4,5), (2,10), and (1,20). For our 
problem we shall suppose that the combination 
(X = 10, D = 2) proves the most practical. Thus, 
with the use of 11 crystals in all and 2 discrim- 
inators, 280 crystal-controlled channels are to be 
obtained. 

1-458. Let ft, fi', f;i . . . . fjsti designate the values 
of f„ from the lowest to the highest, in that order. 
Let fxt, f,;. .... fxtn designate the values f, from 
the lowest to the highest, in that order. The values 
of f„ in ascending order are f,, 2f, .... 7f,.. Fi- 
nally, let frt, and f.i-.. designate the lower and the 
higher discriminator frequencies, respectively. To 
avoid the possibility of spurious conversion fre- 
quencies, the highest value of f,, should be less than 
the minimum associated 1st band-pass amplifier 
frequency, f„ — f„. Also, the lowest value of f, 
should be higher than the highest 2nd band-pass 
amplifier frequency, which, of course, will equal 
the highest discriminator frequency, fd. The order 
of the variable-oscillator frequencies and the se- 
quence of circuit connections required to provide 
each frequency is indicated by the following se- 
quence of equations. 


li 

+ fxl 

— fd 2 100.2 mc 

11 

+ fgl 

— fdi = 100.4 mc 

11 

+ fx. 

— fd 2 = 100.6 mc 

fd =f,. 

+ fx. 

— fd, = 100.8 mc 



etc 

f^., - t 

fxHI 

— fd, = 104 mc 

L, = f. 

+ fxl 

-b fdi = 104.2 mc 

f-' = f. 

+ fxl 

4 - fd-. = 104.4 mc 

f..:, = f,. 

+ fxj 

4- fdi = 104.6 mc 

f^d = f.. 

+ fx-J 

4- fd 2 = 104.8 mc 



etc 

fd„ - f, 

+ fxio 

4- fdj = 108 mc 


fdi - 2 f,. + f„ — fda = 108.2 me 

fd - 2 f,. 4 - L, — fd, = 108.4 me 

f,:! = 2 f.. -h f,a — fda = 108.6 mc 

etc 

f-sn = 7 f,. -f f.m + frtj = 156 mc 

1-459. It can be seen from the equation sequence 
in paragraph 1-458 that the difference between 
the channel frequencies is equal to the difference 


between the discriminator frequencies. Thus, 

Afd = fd 2 — Li = aL = fj — fi = 0.2 mc 

1—459 (1) 

Note in the equation sequence in paragraph 1-458, 
that for a given harmonic frequency, L, all the 
f, crystals are used in sequence before the polarity 
of the discriminator outputs are reversed. In other 
words, fdi and fdo are first subtracted from all com- 
binations of a particular harmonic with the X- 
crystal frequencies and then added to the same 
combinations. This process is repeated with each 
harmonic. If we subtract the equatibn for f| from 
the equation for L, we have 

Afx = f ,2 — f,i = L — fj = 0,4 mc = 2 Afd 

1—459 (2) 

In the general case, 

Afx = D Afd 1—459 (3) 

where D is the number of discriminators. The 
same value of Af, also holds between any other 
two consecutive values of fx. Thus, 

fx2 = fxl + Afx 

f.:i = f„ + 2 Afx 

f,n = fx. + (n - 1) Af, = fx, + (n — 1) D Afd 

1—459 (4) 

The highest frequency of the second crystal oscil- 
lator is given by equation (4) when n = X = the 
total number of crystals. Thus, 

(max) fx = fxl + (X — 1) D Afd 

1_459 (5) 

With the use of equation (6) we can find the low- 
est discriminator frequency, fdi. This is done by 
subtracting the equation for fi>,i from the equation 
for fj,, which gives 

f«i — fat, = fxl — fxKi 4" 2fdi = Afd =- 0.2 mc 

1—459 (6) 

Since f„„ is a particular case of (max) f,, we can 
substitute equation (5) in equation (6) to obtain 
a general equation. We find 

fx. — f.i - (X - 1) D Afd -t 2 fd, = Afd 


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Crystal Oscillators 


On rearranging after canceling out f.i , 


For the fundamental of the 1st crystal oscillator, 


fdi = Afa ~ 1—459 (7) 

The general equation for any particular discrim- 
inator frequency, fa„, is similar to that for f,„ 
given by equation (4) . Thus, , 


fan = fa, + (n - 1) Afa. 1—459 (8) 


and 

(max) fa = fa, + (D - 1) Afa 1—459 (9) 

The next problem is to obtain a general equation 
for fc. This can be had by subtracting the equa- 
tion for fao from the equation for fa,. The re- 
mainder is 

Afa = fo + f,i — f,io — 2 fa- 

where f,,o = (max) f, and faa = (max) fa. Thus, 

f. = 2 fas — f,i -I- f.i + (X — 1) D Afa + Afa 
or 

f„ = 2 fa- + 2 fa. 

fc = 2 [(max) fa + (min) fa] 1 — 459 (10) 

Finally, with fc determined, we can use the equa- 
tion for f, tofindfxi. 

1-450. We are now in a position to express any 
of the circuit frequencies in terms of the param- 
eters fi, Af„, N, H, X, and D. For the nth channel, 

f„ = f, + (n — 1) Af„ 1—460 (1) 

For the highest channel, 

(max) f„ = f, + (N — 1) Af„ 1—460 (2) 

For the lowest discriminator frequency, 

fa, = D) 1_460 (3) 

For the nth discriminator frequency, 
fa„ = At (DX — D + 2n — 1) 

For the highest discriminator frequency, 

At (DX + D — 1) 


t = 2 At D X = 1—460 (6) 

For the nth harmonic frequency, 

t,. = 2 At D X n 1—460 (7) 

For the highest harmonic, 

(max) t — At N 1 — 460 (8) 

For the lowest frequency of the 2nd crystal oscil- 
lator, 

t. = 2f,-At (3DX-D + 1) 

For the nth frequency of the 2nd crystal oscillator, 
„ _ 2 f, — At (3 DX -f D + 1 — 2 D„) 

Ixn 2 

1—460 (10) 

For the highest frequency of the 2nd crystal os- 
cillator. 


(max) t = 


2f, — At (DX + D + 1) 


1—460 (11) 


and the difference between the consecutive values 
of t. 


At - Af„ D 


1—460 (12) 


(max) t 


1—460 (5) 


1-461. On applying the equations in paragraph 
1-460 to the numerical example that has been 
assumed, where f, - 100.2 me, Af, - 0.2 me, 
N = 280, H = 7, X - 10, and D = 2, we find that 

. 0 . 2(1 + 20 — 2 ) 
ti = 2 

tj = 1.9 + 0.2 = 2.1 me 

. 0.2 X 280 . 

I,. = — 8 me 


fi,i. fill’, etc. - 8, 16, 24, 32, 40, 48, 56 me 
f,^g»M^g |0 . -2 + l) ..04.3 me 

At = 0.4 me 

ti.t-, etc = 94.3, 94.7, 95.1, 95.5, 95.9, 96.3, 
96.7, 97.1, 97.5, 97.9 me 


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Crystal Oscillators 


Note that the highest harmonic frequency is equal 
to the bandwidth of the frequency range being 
covered. 

1-462. From the equation, f„ = fh + f, ± fd, it 
can be seen that the frequency stability will be 
approximately the stability of the crystal oscilla- 
tors. This is because f^ is so very much lower than 
fh and f,. Although the discriminators should be 
designed with low-loss, temperature-compensating 
materials, even a large percentage variation in f^ 
would be negligible in its percentage effect upon 
f„. Since it is the sum of fi, and f, that determines 
fo, the maximum percentage frequency deviation 
of the total can be no greater than that of the 
crystal oscillators individually. Without oven con- 
trol, the channel frequencies can be maintained 
within a tolerance of ±0.005 per cent, and better. 

RADIO SET AN/ARC-33 
1-463. Radio Set AN/ARC-33 is an airborne re- 
ceiver-transmitter designed to operate in the v-h-f 
and u-h-f spectrum. This equipment, developed by 
the Bendix Corporation, employs a modified ver- 
sion of the Bendix frequency-synthesizing system 
discussed in the foregoing paragraphs. The fre- 
quency-control section (see figure 1-192 for block 
diagram) is designed to permit receive-transmit 
communication on any one of 1750 channels 
spaced 100 kc apart in the 226-to-399.9-mc band. 
1-464. An important modification in the synthe- 
sizing system arises from the fact that the syn- 
thesized frequencies are utilized as heterodyne in- 
jection signals during reception and as carrier 
signals during transmission. The tuning controls 
are such that the receiver and transmitter circuits 
are always automatically tuned to the same chan- 
nel, but since the desired receiver injection fre- 
quency must differ from the tuned channel fre- 
quency by an amount equal to the receiver inter- 
mediate frequency, the design engineers had to 
decide whether to let the variable-frequency oscil- 
lator be, in effect, a subharmonic local oscillator 
for the receiver or a subharmonic master oscilla- 
tor for the transmitter. They decided in favor of 
the receiver. Thus, the stabilized output frequency, 
f„, of the vfo, after being multiplied 12 times, is 
used directly as the injection voltage in the 1st 
mixer stage in the receiver. This provides a fixed 
intermediate frequency, f,, of 15.325 me for each 
of 1750 channels. Now, the v-f-o output, fo, is also 
used in the synthesis of the channel frequency of 
the transmitter. Since the 12th harmonic of f„ 
always differs from the channel frequency by an 
amount equal to f,, all that needs to be done in 
principle is to mix 12f„ with a fixed oscillator fre- 


quency equal to fi and for the sum product to be 
isolated and amplified for use as the transmitter 
carrier. In the ARC-33 transceiver this effect is 
achieved by mixing the output, f„ of a crystal- 
controlled oscillator (called the “sidestep” oscil- 
lator) with the 6th harmonic of f„, then selecting 
the sum frequency (6f„ plus f.) and doubling it to 
form the carrier frequency, f., which, after ampli- 
fication, is fed to the antenna. Summarizing these 
frequency relations in the form of equations, we 
have: 

injection frequency = 12 f^ 

intermediate frequency = f, = f, — 12fo 

antenna (channel) frequency = f, = 12f„ + fi 
= 2 (6fo + f.) 

sidestep frequency = f, = fi/2 

Note the necessary harmonic relation between the 
sidestep output and the intermediate frequency. It 
is also important to note that the principle in- 
volved in the use of a sidestep oscillator permits, 
not only a Bendix synthesizing system, but any 
synthesizing system to be readily modified for the 
dual-purpose requirements of transceiver fre- 
quency control. 

1-465. Comparison of figures 1-191 and 1-192 will 
reveal that the 2nd crystal oscillator in the basic 
Bendix synthesizer has been replaced by two 
crystal oscillators (the 2nd and 3rd in figure 
1-192) in Radio Set AN/ARC-33. This has been 
done to permit a greater number of frequencies 
with a fewer number of crystals. 

1-466. Another significant modification occurs in 
the ARC-33 discriminator circuit. As is explained 
in more detail in a subsequent paragraph, the 
ARC-33 discriminator is not a conventional type 
that employs a parallel tuned circuit to control the 
phase differences between the input voltage com- 
ponents. In that type of discriminator the output 
voltage always has a net d-c component unless the 
input frequency is equal to the antiresonant fre- 
quency of the tuned tank. The polarity of the d-c 
% component is an index of whether the input fre- 
quency is higher or lower than that at which the 
tank is tuned, and the amplitude of the d-c com- 
ponent can be a measure of the amount of differ- 
ence between the two frequencies. It is this type 
of discriminator that is assumed in the discussion 
of the basic Bendix synthesizing system, the d-c 
output of which is used to control the variable 
oscillator tuning. In the ARC-33, however, the 
discriminator does not employ a tuned circuit, but 
instead is fed a signal that is controlled by the 4th 
crystal oscillator. The discriminator also receives 


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Figure 1-192. Block diagram of frequency control section of Radio Set AN/ARC-33 showing modified version of Bendix synthesizing system 




































Saclien I 

Crystal Oscillators 


an input signal from the 4th mixer. The two sig- 
nals combine in the discriminator Circuit to pro- 
vide a net d-c output only when the two signals 
are of the same frequency. When the frequencies 
are identical, the behavior of the discriminator is 
quite similar to the behavior of one that employs 
a tuned circuit, the d-c output depending upon the 
differences in phase between the input voltages. 
Since the 4th oscillator can be controlled by either 
one of two crystals, this arrangement is equiva- 
lent to having two discriminators of the tuned- 
circuit type. 

1-467. In paragraph 1-456 it is explained that for 
each discriminator, two values of f„ are possible. 
In Radio Set AN/ARC-33 only one of these values 
is used for each 4th-oscillator frequency, namely 
fo = fh + f. + fd 1-467 (1) 
In the equation above, ft, is the selected effective 
harmonic of the 1st crystal oscillator, equal to fm 
on the low band and to (fhi -f- fi, 2 ) on the high 
band, f, equals the sum of the frequencies of the 
2nd and 3rd crystal oscillators (f ,2 + f.s), and fa 


is the frequency of the 4th crystal oscillator fed 
to the discriminator. 

1-468. Other modifications of the Bendix system 
as occur in the frequency-control circuits of Radio 
Set AN/ARC-33 are of an even less radical nature 
than those described above. There is the division 
of the 1st bandpass stage into low-band and high- 
band circuits, and there is the addition of an a-f-c 
reactance tube, which is actually more of an ex- 
tension of the modification caused by the use of a 
crystal-controlled discriminator, but these and 
other special circuit arrangements are best ex- 
plained in the more detailed analyses later. At 
this point it will be helpful to examine briefly the 
role each of the various oscillator frequencies 
plays in controlling the final antenna frequency. 
1-469. First, we shall examine the simplified block 
diagram shown in figure 1-193. The frequency- 
control system indicated represents an imaginary 
synthesizing circuit that provides the same trans- 
mitter output frequencies, f», as does Radio Set 
AN/ARC-33. The principal difference between the 


TO TUNSWTTER 



Figure 7-193. Simplified block diagram of the frequency-conlrol section of Radio Set AN/ARC-33 at it would 
appear if the variable frequency otcillator were the direct generator of the transmitter output frequency with- 
out the use of multiplier or side-step circuits 


WADC TR 56-156 


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imaginary and the actual systems is that the 
imaginary system is hot required to provide an 
injection voltage for a receiver heterodyne circuit, 
nor is it required to employ multiplier stages fol- 
lowing the variable oscillator. In other words, f„ 
can be assumed to equal f„. Under these conditions 
the various crystal and harmonic frequencies 
would assume the simple values shown in figure 
1-193. The actual circuit frequencies are those in- 
dicated in the frequency diagram of figure 1-194. 
A comparison of the frequencies in the two sys- 
tems will show that the injected frequencies in 
each mixer stage of the imaginary system vary 
from one to the next in steps that are 12 times 
greater than are the corresponding steps in the 
actual system. This does not mean, except in the 
case of the 1st crystal oscillator and its harmonics, 
that the ima^nary crystal frequencies are 12 
times the actual crystal frequencies ; it is only the 
differences between adjacent frequencies that are 
related in the proportion of 12 to 1. Since the fre- 
quency of the variable frequency oscillator in the 
actual circuit is eventually multiplied 12 times, it 
can be seen that the frequency steps in the actual 
circuit are equivalent to those in the imaginary 
circuit insofar as they add or subtract in the con- 
trol of the antenna frequency. Thus, we can say 
that the antenna frequency is effectively synthe- 
sized in 10-mc units by the Ist crystal oscillator 
and harmonic generator, 2-mc units by the 2nd 
crystal oscillator, 0.2-mc units by the 3rd crystal 
oscillator, and finally to the nearest 0.1-mc unit 
by the discriminator and 4th crystal oscillator. 

Detailed Circuit Description 
1-470. The principal component of Radio Set AN/ 
ARC-33 is Receiver-Transmitter RT-173/ ARC-33. 
The receiver-transmitter is divided into a number 
of sectional components, two of which are of im- 
portance to us : 

a. The monitor chassis, which contains all the 
crystal circuits for controling the variable- 
frequency oscillator. 

b. The r-f head, which contains the variable- 
frequency oscillator, the multiplier circuits, the 
1st i-f mixer, the sidestep oscillator, as well as 
the r-f amplifiers of the receiver and the power 
amplifiers of the transmitter. Also of importance 
to us are the relays which control the tuning 
motor. These are mounted on the main frame. We 
shall discuss the monitor circuits first, and then 
those in the r-f head. Except for occasional in- 
sertions and editing, the descriptions to follow 
are largely extracts from USAF Technical Order 
No. 12R2.2ARC33-2. 


Section i 
Crystal Oscillotors 

Monitor Chassis 

1-471. The monitor chassis in Radio Set AN/ 
, ARC-33 concerns only the frequency control of 
the variable-frequency oscillator and electronic 
control of the tuning-capacitor drive motor with 
its clutches. There are no other circuits involved. 

The detailed circuit descriptions of the mon- 
itor chassis are made with reference to the com- 
ponent symbols employed in the block diagram of 
figure 1-195 and the schematic diagram of figure 

1-196. With the exception of a coaxial connector 
for the r-f input from the variable-frequency 
oscillator, all external connections to the unit are 
made through a single connector, which is so 
arranged that connection automatically is made 
when the chassis is inserted in its proper place in 
the main frame. 

1-472. FIRST CRYSTAL OSCILLATOR. The 1st 
crystal oscillator is a single-frequency, funda- 
mental-mode, 833.333-kc oscillator ol the cathode- 
coupled Butler type. A selected harmonic of the 
oscillator is mixed with the frequency of the vari- 
able frequency oscillator in the 1st mixer. A dual 
triode tube, V401, is employed as the oscillator 
tube, which has two output connections. Section A 
is tuned to the 5th harmonic of the crystal and 
feeds the grid of frequency multiplier V403B. 
The plate output of the grounded-grid oscillator 
section B is coupled through capacitor C407 to 
the harmonic generator grid. The crystal unit, 
which is of the type CR-28/U, is mounted in a 
type HD-54/U crystal oven. The oven employs 
two heaters and thermostats, one heater being 
used to bring the temperature quickly up to the 
operating level, whereas the other, which has a 
lower wattage, is used to maintain constant oper- 
ating temperature. In order to check the oscillator 
for proper operation, a test connection for meas- 
uring rectified grid current is brought out to test 
socket X412. The stability of the final transmitter 
frequency is more dependent upon the stability 
of this oscillator than upon that of any of the 
others. The reason is that the 1st crystal oscillator 
controls a greater percentage of the final fre- 
quency, especially in the high band, than do the 
other oscillators. This can readily be seen if we 
visualize the final frequency as being synthesized 
by adding together the crystal-oscillator frequen- 
cies in the simplified block diagram of figure 1-193. 
The key function of this oscillator is the reason 
why the highly stable Butler circuit is employed. 

1-473. HARMONIC GENERATOR. The function 
of harmonic generator V403A is to produce any 
selected harmonic of the first crystal oscillator 


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Section I 

Crystal Oscillators 



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Figuro 1-194. Frequency diagram of Frequency-confro/ sysfem in Radio Set AN/ARC-33 



























Sech'on I 
Crystal Oscillators 



Figure I-f95. Monitor chassis. Block diagram of a~f-e system in Radio Set AN/AftC-33 


output from the 4th to the 13th, inclusive. The 
selected harmonic is fed to the 1st mixer to be 
heterodyned with the fi-equency from the vari- 
able frequency oscillator. Harmonic selection is 
achieved by capacitance tuning the primary and 
secondary of the harmonic generator output 
transformer T401 to the desired harmonic fre- 
quency. The switching is accomplished by 10- 
position rotary switches S403A and S403B, 
which are driven by the selector motor through 
a harmonic generator clutch. These switches 
merely select the proper fixed capacitors for tun- 
ing the primary and scKiondary winding to the 
desired harmonic of the' 833.333-kc fundamental. 
1-474. R-f input from the 1st crystal oscillator is 
fed to the harmonic generator tube, V403A, 
thi-ough coupling capacitor C407. A grid bias far 
below cutoff is piovided by grid resistor R407 in 
order to ensure an output rich in harmonics. The 
primary and secondary windings of transformer 


T401 are permeability tuned for alignment at the 
lowest (4th) harmonic. The highest frequency, 
which is the 13th harmonic, is determined by 
capacitoi s C418 and C419 across the primary and 
C42fl and C421 across the .secondary. Capacitors 
C418 and C421 are trimmers for alignment at 
the highest frequency. The .selection of all har- 
monics, up to but not including the 13th, is 
accomplished by switching in the proper fixed 
capacitor C U)8 through C417 across the primary 
of ti-ansfoimer TlOl, and C222 through C431 
across the secondary. For the lowest harmonic, 
C416 and Cl 17 for the primary and C430 and 
C431 for the secondary are connected in parallel. 
For the highest harmonic, no auxiliarv capacitor 
is switched into the circuit. For proper operation 
of the monitor, it is necessary that the input level 
of the 1st mixer be approximately the same for 
each selected harmonic. This is accomplished by 
selecting a grid bias for the harmonic generator 


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Section I 

Crystal Oscillators 


which causes the output level to vary inversely 
with frequency under a constant plate load im- 
pedance. Then, by maintaining the “Q” of the 
transformer T401 windings constant at all se- 
lected frequencies, the resultant plate load im- 
pedance of tube V403 can be made to vary directly 
with frequency because of the increasing induc- 
tance/capacitance ratio. The resultant voltage 
across the T401 transformer windings, therefore, 
is essentially constant regardless of the harmonic 
selected. 

1-475. FIRST MIXER. The 1st mixer tube, V402, 
combines the output of the harmonic generator 
with the output frequency of the variable fre- 
quency oscillator (vfo) whose frequency is to be 
controlled by the monitor circuits. The 1st mixer 
output consists of the difference frequency be- 
tween the selected harmonic generator frequency 
and the v-f-o frequency, the latter frc^jnency al- 
ways being the higher. Input from the harmonic 
generator is fed directly to the control grid, 
whereas the input from the vfo, which enters the 
monitor chassis through connector J401, is fed 
from isolation amplifier V402A to the cathode of 
the 1st mixer through coupling capacitor C432. 
Output coupling is provided by bandpass coil 
assembly Z401, which consists of a low-band cir- 
cuit and a high-band circuit. The bandpass selec- 
tor relay selects the proper circuit of bandpass 
coil assembly Z401. The reason for using two 
bandpass circuits is that the broad frequency 
range of the radio set makes it necessary to divide 
the range into two smaller ranges. The band- 
determining factor is the selection of the 1st 
digit of the desired channel frequency at the con- 
trol panel. The control pane! switch energizes 
the band selector relay when the 1st digit is 2; 
that is, when the antenna frequency is to be less 
than 300 me. For antenna frequencies of 300 me 
and above, the band relay is unenergized. Coil 
assembly Z401 is designed to pass a band from 
approximately 12.1 to 12.9 me and a band from 
approximately 20.4 to 21.2 me. Because of the use 
of two bandpasses at the 1st mixer output, each 
of the harmonic generator selected output fre- 
quencies is used twice, once in each band. 

1-476. BANDPASS AMPLIFIER. The bandpass 
amplifier, V404, is an amplifier for the 1st mixer 
output on the low-frequency band and is used to 
improve the bandpass characteristics of the cir- 
cuit. This stage is unconventional inasmuch as it 
also functions as a 2nd mixer, the operation of 
which is described in the following paragraph. 
The plate load consists of transformer T402, 
which is tuned to the same frequency band as the 


low band of Z401. The bandpass amplifier is 
capacitance-coupled through capacitor C443 to 
the number one grid of the 3rd mixer, V405. 
1-477. SECOND MIXER. The 2nd mixer stage 
utilizes the same tube elem.ents of V404 as are 
used when the tube is operated as a bandpass 
amplifier. However, this dual usage does not occur 
simultaneously. Tube V404 functions as a 2nd 
mixer only when the radio set operates in the 
high band of the frequency range. The 2nd mixer 
combines the 1st mixer output frequency with the 
10th harmonic of the 1st crystal oscillator fre- 
quency. The output consists of the difference 
frequency, where the 10th harmonic frequency is 
always lower than the output frequency of the 
1st mixer. Note that the reduction of the 1st 
mixer output frequency by an amount equal to 
the 10th harmonic of the 1st crystal oscillator 
frequency is equivalent to extending the harmonic 
generator range from the 13th to the 23rd har- 
monic and eliminating the high band of bandpass 
coil assembly Z401. Input to the 2nd mixer from 
the high-band bandpass filter is fed to the control 
grid along with the input from frequency multi- 
plier V403B. The plate load consists of trans- 
foi'mer T402, which is tuned to pass a band from 
12.1 to 12.9 me. The desired band width for the 
transformer is obtained through the use of load- 
ing resistors R418 and R419. Transformer T402 
is the same plate load circuit for the 2nd mixer 
as is used when the stage operates as a bandpass 
amplifier. 

1-478. FREQUENCY MULTIPLIER. The fre- 
quency multiplier, V403B, is fed with the 5th 
harmonic output from the 1st crystal oscillator. 
The output transformer, T406, is tuned to 8.33333 
me, which is the 2nd harmonic of the multiplier 
input frequency. Thus, V403B is a frequency 
doubler and feeds its output, the 10th harmonic 
of the 1st crystal oscillator frequency, to the 
control grid of the 2nd mixer, V404. When the 
radio set is operated in the low band of its fre- 
quency range, no output is derived from the fre- 
quency multiplier because plate voltage is removed 
from tube V403B. The application of plate voltage 
is controlled by the band selector relay. When the 
band selector relay switches the high bandpass 
filter into the 1st mixer circuit, it also applies 
voltage to the plate of the frequency multiplier 
so that an 8.33333-mc signal is fed to the 2nd 
mixer. The frequency multiplier is capacitively 
coupled to the control grid of the 2nd mixer 
through capacitor C457. 

1-479. SECOND CRYSTAL OSCILLATOR. The 
2nd ci'ystal oscillator provides a selection of any 


WAOC TR 56-156 


304 




WADC TR 56-156 























Section I 
Crystal Oscillators 


4TH MIXER 
JAN 572S/6ASEV 
V407 
X407 


DISCRIMINATOR 
JAN 57E6/ 6N.5W 
V409 



TEST SOCKET 
X4I2 

(TOP VIEW) 


PIN I aUTCN CONTROl VOITAGE 

2 +200V TEST 

3 +I05V REG TEST 

4 1ST OSC TEST 

5 2 NO OSC TEST 

6 3R0 OSC TEST 
T 4TH OSC TEST 

8 GROUND 

9 DISC OUTPUT 
K) GROUND 
II DISC TEST 


0 409 


(E3400M^ 
(4 3233333 ) (D3 566667 ) 
(D3 900000 ') (C3.733333 ) 


MONITOR OVENS AN/ARC'33 
CRYSTAL PLACEMENT 
0410 



(S 5. 181250 ) 

C F3.650000() 

(R5.I729I7 ) 

( C3733333 ) 

(0 3 750000 ) 

(h not used ) 

CP3 666667 ) 

(JS 716667 ) 

(03 766667 ) 

(K3 800000) 

(N3.683333 ) 

(L 3 700000) 

(M3 783333 ) 



csobJ, 

TOOOll \ | 7000 [/a^/^ 


_C5I0 

2000 


C5lll 
2000 \ 


■ 


HES3 

■ 


Mi'MI 


Tc5I4 -4- 
J 7000 


UNDERSIDE VIEW 
OF SEALED 
RELAYS 


d) ® C®®® CD 


(j!)® (iD @ I 


note: 

ALL RESISTANCE IN OHMS 
ALL CAPACITANCE IN MMF 
UNLESS OTHERWISE STATED. 
X^IOOO 


COLOR 



a 


E!] 

BLACK 

m 

Miai 

a 

GREEN 

n 

HIM 

a 

RED 

m 

SLATE 

a 

VIOLET 

□1 

WHITE 

ai 

YELLOW 


S3 


Figure 1-196. Monitor chassis. Schematic diagram of a-f-c 
system in Radio Set AN/ARC-33 


o 


305-306 













one of five crystal-controlled frequencies by means 
of crystal switching. The crystal-controlled out- 
put frequency is' fed to the suppressor grid of the 
3rd mixer to be mixed with either the 2nd mixer 
or the bandpass amplifier output frequency. The 
oscillator circuit utilizes one half of the duplex 
triode tube V406 as a grounded-plate Pierce 
oscillator. R-f output is taken from the cathode. 
Each of the five crystal units, which are of the 
CR-27/U type, has a separate trimmer so that 
each selected 2 nd crystal oscillator frequency can 
be adjusted exactly. All of the crystal units with 
their trimmers are enclosed in a single oven 
(Bendix Radio type L205628) which is kept at 
approximately 75°C (167°F) by a thermostat- 
controlled heater. A booster heater and thermo- 
stat are provided in addition for quick warmup. 
In order that the oscillator operation may be 
checked, a test connection from the grid circuit 
is brought out for checking rectified grid current 
at test socket X412. 

1-480. THIRD MIXER. The circuit of the 3rd 
mixer, V405, is similar to that of the 2nd mixer 
except that the signal voltage from its heterodyne 
crystal oscillator is injected at the suppressor 
grid. The 3rd mixer output circuit is tuned to a 
center frequency of 8.9 me and is designed to pass 
a band approximately from 8.8 to 9.0 me. The 3rd 
mixer combines the output frequencies of either 
the 2 nd mixer or bandpass amplifier and the 2 nd 
crystal oscillator, the output frequency of the 
former two always being the higher. The 3rd 
mixer output, which is the difference frequency, 
is fed from the secondary of transformer T403 
to the 4th mixer through capacitor C449. 

1-481. THIRD CRYSTAL OSCILLATOR. The 
3rd crystal oscillator, the V406B circuit, is of the 
same design as the 2 nd crystal oscillator except 
for the operating frequencies. The crystal units, 
which are of the CR-27/U type, are divided into 
two groups of five each. All are used throughout 
the total frequency range of the radio set. The 
3rd crystal oscillator employs the second half of 
the same tube that is used for the 2 nd crystal 
oscillator. The 10 crystal units with their trim- 
mers are housed in a 13-position oven (Bendix 
Radio type N205651), of which one position is 
not used. The other two positions are used to 
mount the two crystals of the 4th crystal oscil- 
lator. The oven is thermostatically controlled at 
75°C (167°F). A separate booster heater and 
thermostat are provided for quick warmup. 

1-482. FOURTH MIXER. The 4th mixer, V407, 
is identical to the 3rd mixer and operates in the 
same manner. The 4th mixer output circuit is 


Section I 
Crystal Oscillators 

designed to pass a band of approximately 5.1 to 
5.2 me. In the 4th mixer are combined the output 
frequency of the 3rd mixer and that of the 3rd 
crystal oscillator, the former frequency always 
being the higher. The difference frequency is 
selected by the tuned plate transformer and is 
inductively coupled into the discriminator circuit. 
1-483. DISCRIMINATOR. The purpose of the 
discriminator, V409, is to indicate a deviation in 
phase or frequency between its two inputs. One 
of these inputs is obtained from the 4th crystal 
oscillator and is used as the reference signal. The 
other discriminator input is derived from the 4th 
mixer output, and it is the deviation in phase of 
this signal from that of the reference signal that 
is to be indicated by means of the output voltage 
across R430 and R431. The magnitude of this 
discriminator voltage indicates the extent of the 
deviation, whereas the polarity indicates the di- 
rection of the deviation. The circuit design is 
similar to the type employed for frequency- 
modulation (FM) receivers, except that in this 
case the reference voltage is obtained from a 
separate reference oscillator instead of a parallel 
resonant tank circuit. The voltages applied to the 
discriminator are shown in figure 1-197. E, is the 
input from the 4th mixer and Ek is the reference 
voltage from the 4th oscillator. To simplify the 
discriminator explanation, Er, insofar as it adds 
vectorially with Ee to form the r-f voltages across 
the two diodes, is best interpreted in terms of two 
equal and separate voltages 180 degrees out of 
phase. One is E;;n, the r-f voltage at pin 2 with 
respect to point B; the other is Etei, the r-f volt- 
age at pin 7 with respect to point B. Assuming 
that the bypass capacitors C452 and C453 offer 
zero impedances to the r-f signals, pins 1 and 5 
and point A are all at r-f ground potential. Thus, 
the r-f voltage (E:;) of plate pin 2 with respect 
to cathode pin 5 is the same as the voltage of 
pin 2 with respect to point A. Similarly, E 7 , the 
r-f plate voltage of the second diode, is equal to 
the voltage of pin 7 with respect to point A. Now, 
Ei and E 7 have one voltage component in common, 
which is El, the voltage of point B with respect 
to point A. Thus, Ea and E 7 are equal to the re- 
sultants, respectively, obtained by adding vector- 
ially to the common voltage E, the voltages E^ei 
and E 7 E 1 . If we assume that C452 charges to the 
peak of Ea and that C453 charges to the peak of 
E7, the d-c polarities will be as indicated in figure 
1-197 (D), where point A is shown as negative 
with respect to the two cathodes. The d-c output 
equals (Ea — E7), where both voltage symbols 
represent the positive peak magnitudes only. The 


WADC TR 56-156 


307 



Section I 

Crystal Oscillators 


output is positive or negative according to 
whether Ea is greater or less than E 7 , respec- 
tively; and this is dependent, respectively, upon 
whether E, lags or leads the zero-output position, 
which is the 90-degree phase displacement from 
E„ that is shown in figure 1-197 (A). Upon exam- 
ination of the discriminator diagram it can be 
seen that the application of either of the input 
voltages alone does not develop a discriminator 
output, since in each case E 2 and E 7 will equal 
each other, and hence equal currents will flow in 
opposite directions through the two halves of dis- 
criminator load R430 and R431. Also, it can be 
seen from the vector diagram (A) that, if the 
two discriminator input voltages are exactly 90 
degrees out of phase, the resultant voltages ap- 
plied to the diode plates are equal, thus pi'oducing 


zero discriminatoi- output. If, however, the vari- 
able frequency oscillator should vary in phase, the 
resultant voltages applied to the discriminator 
plates are unequal, as shown in vector diagrams 
(B) and (C). This results in a discriminator out- 
put voltage whose value and polarity depend upon 
the magnitude and direction, respectively, of the 
phase deviation. If there is a frequency deviation, 
the phase of the 4th mixer output, E,, rotates 
completely around the 4th crystal oscillator out- 
put, Er, with a consequent a-c voltage appearing 
in the discriminator output. The frequency of 
this ac is equal to the difference between the two 
discriminator input frequencies. 

1-484. Discriminator output is applied to the 
tuning-motor-control amplifier and to the reac- 


PIN 7 

FOURTH MIXER OUTPUT 
E] 90' OUT OF PHASE 
WITH FOURTH CRYSTAL 
VOLTAGE Er- 
DISCRIMINATOR OUTPUT* 
ZERO 

|MAX E 2 I-IMAX E 7 I* 

ZERO D-C OUTPUT. 



A 


PIN 2 


(R-F GROUND) 


(A) 


PIN 

fourth mixer OUTPUT 
E] LAGGING NORMAL 
PHASE. INDICATING THAT 

v-F -0 frequency is 

TENDING TO GO LOW. 
RESULTING POSITIVE 
DISCRIMINATOR OUTPUT 
WILL CAUSE MOTOR TO 
TUNE VFO HIGHER IN 
FREQUENCY. 

|MAX E 2 I -|MAX E 7 I* 
POSITIVE 0-C OUTPUT. 



A 


(R-F GROUND) 
(B) 


PIN 7 

FOURTH MIXER OUTPUT 
E| LEADING NORMAL PHASE, 
INDICATING THAT V-F-O 
FREQUENCY IS TENDING 
TO GO HIGH. 

RESULTING NEGATIVE 
DISCRIMINATOR OUTPUT WILL 
CAUSE MOTOR TO TUNE 
VFO LOWER IN FREQUENCY. 
}MAX E 2 I - (MAX E 7 I • 
NEGATIVE 0-C OUTPUT. 



GROUND) 


OUTPUT TO 
REACTANCE 
TUBE AND MOTOR 

DISCRIMINATOR CONTROL AMPLIFIER 



(D) 


Ej* INPUT FROM 4TH MIXER 

Er» REFERENCE VOLTAGE FROM FOURTH CRYSTAL OSOLLATOR 

• ±(^28 + E7b)“±(E2B “£ 73 ) 

Eg -RESULTANT VOLTAGE APPLIED TO DIODE PLATE 
(SOCKET PIN 2) 

E 7 -RESULTANT VOLTAGE APPLIED TO DIODE PLATE 
(SOCKET PIN 7) 

K- IN ACTUAL PRACTICE THE V-F-O FREQUENCY CONTROL 
CIRCUITS ARE SUCH THAT "ON FREQUENCY" 

DISCRIMINATOR OUTPUT IS APPROX +5 VOLTS. 

THUS. IN NORMAL OPERATION, VECTOR Ej WILL BE 
TILTED SLIGHTLY TO THE RIGHT OF ITS 90* POSITION 
SHOWN IN FIGURE (A) AT LEFT. 

DUE TO THE ACTION OF THE DIODES AND THEIR LOADS. 
THE DISCRIMINATOR RESULTANT 0-C OUTPUT IS EQUAL 
TO THE DIFFERENCE IN MAGNITUDE BETWEEN Eg AND 
E 7 ({MAX Egl-jMAX E 7 I). 


(C) 


Figure 1-197. Discriminafor operation as a function of the phase difference be/ween the input voltages 
WADC TR 56-156 308 




Section I 
Crystal Oscillators 



tance tube shunted across the variable frequency 
oscillator. When the variable frequency oscillator 
approaches the desired operating frequency, a 
point is reached when the discriminator output 
frequency becomes equal to or less than the maxi- 
mum frequency at which the reactance tube can 
respond. At this point the reactance tube imme- 
diately locks the variable frequency oscillator 
exactly to its correct frequency. When this occurs, 
the discriminator output is dc and is proportional 
to the phase difference between its two input 
voltages, which difference is, in turn, proportional 
to the amount of “pull” exerted by the reactance 
tube. This d-c control voltage is fed to the motor- 
control amplifier, V411, and causes the tuning 
motor to drive the variable frequency oscillator 
tuning capacitor to that point which eliminates 
excessive pull by the reactance tube. The circuit 
design is such that the equilibrium point corre- 
sponds to a discriminator output of approximately 
5 volts positive. Normally it would be assumed 
that the “on-frequency” point would be reached 
when the discriminator output dropped to zero, 
which would occur when the two input signals 
were exactly 90 degrees out of phase. The zero- 
voltage state is not used, however, in order to 
avoid ambiguity in identifying the on-frequency 
condition and to simplify the control circuit by 
having an equilibrium control voltage of definite 
magnitude. For example, zero output not only 
occurs at each of the two (plus and minus) 90- 
degree phase conditions, but also occurs when the 
variable frequency oscillator is so far off fre- 
quency that there is no input from the 4th mixer, 
or if one or both of the input voltages fails, or 
when the two input frequencies are different and 
the discriminator records the best frequency. 
Thus, a plus 5-volt reference is used, and the 
control circuits are so biased that this control 
level establishes the on-frequency condition. The 
ac which the discriminator develops before the 
reactance tube “pull-in” point is reached, is pre- 
vented from affecting the motor-control amplifier 
by the low-pass resistance-capacitance filter, 
R451 and C445A. 

1-485. By employing a phase-sensitive discrim- 
inator, it is possible to feed a correction voltage 
to the control circuits before an actual frequency 
deviation occurs, since a frequency deviation, un- 
less it is an instantaneous, discontinuous jump, is 
first indicated as a phase deviation. Therefore, 
no frequency error occurs except that which may 
be due to the reference crystals themselves. All 
small and rapid frequency shifts of the variable 
frequency oscillator are corrected by the reactance 


tube. Larger and slower drifts are corrected by 
the motor-control amplifier and the tuning motor. 
An extremely large and sudden frequency jump 
of the variable frequency oscillator which takes it 
out of the range of the reactance tube causes the 
entire tuning sequence to recycle. However, this 
does not occur during normal operation. To aid 
in discriminator alignment and test, two test 
points are brought out to pins of test socket X412. 
One of these, pin 9, makes it possible to measure 
the total discriminator output. The other, pin 11, 
is connected to the load center-tap for checking 
the discriminator operation. 

1-486. FOURTH CRYSTAL OSCILLATOR. The 
4th crystal oscillator is used to control the oper- 
ating frequency of the discriminator and thereby 
has final control of the exact frequency of the 
variable frequency oscillator. The 4th crystal 
oscillator can be switched to either one of two 
type CR-27/U crystal units, whose frequencies 
are spaced 8.333 kc apart. This spacing is equal 
to one-twelfth of the channel spacing of 100 kc. 
Thus, it is the spacing of the 4th crystal oscillator 
frequencies that determines the channel spacing. 
For maximum stability the crystal units are 
mounted in the same crystal oven that houses 
the crystals of the 3rd crystal oscillator. Crystal 
selection is controlled by a selector switch on the 
main control panel. The selection is determined 
by the choice of the 4th digit of the channel 
frequency. 

1-487. The 4th crystal oscillator circuit employs 
one half of a duplex triode tube, V410, which is 
connected as a radio-frequency grounded-plate 
Pierce oscillator, with the circuit designed in a 
manner similar to those of the 2nd and 3rd crystal 
oscillators. The output is taken from the cathode 
and coupled through C471 to the grid of the other 
half of tube V410, which is operated as a cathode 
follower. The output of the cathode follower is 
inductively coupled into the discriminator circuit 
through transformer T405. The purpose of the 
cathode follower is to isolate the loading effects 
of the discriminator from the 4th crystal oscil- 
lator. As a check on the 4th crystal oscillator 
operation, a test connection for rectified grid cur- 
rent measurement is brought out to test socket 
X412, pin 7. 

1-488. CLUTCH CONTROL TUBE. The purpose 
of the clutch control tube V408, is to shift the 
r-f head tuning drive from the medium-speed 
clutch to the low-speed clutch for fine tuning of 
the exact channel frequency. Whenever the fre- 
quency of the variable frequency oscillator, after 


WADC TR 56-156 


309 



Section I 

Crystal Oscillators 


being fed through the four mixers, comes within 
range of the discriminator input tuning at trans- 
former T404, the tuning should be shifted into 
low speed. Thus, by coupling part of the discrim- 
inator input through capacitor C464 to clutch- 
control rectifier V408A and amplifying the recti- 
fier output through V408B, a control voltage is 
obtained which is used ito operate the clutch- 
control relay. The clutch-control relay is mounted 
on the main frame and has its coil connected in 
series with the plate of clutch-control amplifier 
V408B. The relay is energized when the tube grid 
receives no excitation. When a signal is fed to the 
discriminator from the 4th mixer, a portion of 
the signal is fed through capacitor C464 and 
rectified in diode-connected tube V408A. The re- 
sulting current flow in resistor R443 causes the 
ungrounded end to become more negative. This 
negative voltage is fed to the grid of the clutch- 
control amplifier through low-pass filter R446 
and C463, cutting the tube off. This in turn 
de-energizes the clutch-control relay and places 
the low-speed clutch in operation. It should be 
noted here that the over-all tuning range of the 
radio set is divided into two nearly equal bands 
and that while the variable frequency oscillator 
tuning is being driven through the unused band 
a spurious frequency may pass through to the 
discriminator and the clutch-control tube. In this 
case, however, even though the clutch-control 
relay is de-energized, the tuning remains in high 
speed because of a lockup circuit. 

1-489. MOTOR-CONTROL AMPLIFIER. The 
motor-control amplifier, V411, is a d-c amplifier 
whose purpose is the control of a reversible tun- 
ing motor in accordance with a d-c control voltage 
from the discriminator. The discriminator output 
voltage is fed to the grid through the low-pass 
resistance-capacitance filter R451 and C445A so 
that the ac, which appears in the discriminator 
output as the channel frequency is being tuned, 
does not influence the operation of the amplifier. 

As its plate load, the motor-control amplifier 
works into two relay coils in series. These are 
called the high and low discriminator relays. The 
two relays are mounted on the main frame. The 
high discriminator relay is designed to “pull in” 

- . transmitter output 

v-f-o frequency = 

The screen grid of the v-f-o tube, V802, is the 
anode of the oscillatory circuit. The plate circuit 
is tuned to three times the grid frequency, thereby 
tripling in this tube. Coupling capacitor C907 
from grid inductor 1801 feeds a small amount of 


at approximately 9 ma ^nd to “drop out” at ap- 
proximately 6 ma. The low discriminator relay is 
designed to pull in at approximately 5 ma and to 
drop out at approximately 3 ma. The plate current 
of the motor-control amplifier is so adjusted that, 
with a plus 5-volt “on-frequency” output from the 
discriminator, approximately 5.5 ma plate cur- 
rent flows. This is sufficient to energize the low 
discriminator relay but not the high one. The 
contacts of the low discriminator relay are so con- 
nected that when the relay is energized the tuning 
motor (mounted in the r-f head) is de-energized. 
If the frequency of the variable frequency oscil- 
lator should tend to drift too low, the discrimina- 
tor voltage controlling the amplifier becomes more 
positive, increasing the amplifier plate current 
and thereby energizing both relays. This has the 
effect of causing the tuning motor to turn in a 
direction that diminishes the tuning capacitance 
and hence raises the frequency. If the variable 
frequency should tend to drift too high, the plate 
current of the motor-control amplifier decreases 
to a value below 3 ma and both relays are de- 
energized. This has the effect of causing the 
tuning motor to turn in the opposite direction, so 
that the frequency of the variable frequency oscil- 
lator is decreased. Note that in order to keep the 
motor-control amplifier bias constant, regardless 
of plate current flow, a positive 6-volt cathode 
bias is obtained from the d-c drop across the 
heater, rather than from the drop across a cath- 
ode biasing resistor. 

R-F Head 

1-490. The r-f head contains all the main channel 
transmitter and receiver r-f circuits including the 
reactance tube, the tuning capacitor drive mech- 
anism, and certain tuning control circuits. A block 
diagram is shown in figure 1-198 and a schematic 
diagram is shown in figure 1-199. Beginning with 
the variable frequency oscillator, we shall discuss 
first the transmitter circuits and then the re- 
ceiver circuits. 

1-491. VARIABLE FREQUENCY OSCILLATOR 
AND TRIPLER. The variable frequency oscilla- 
tor employs an electron-coupled Hartley circuit. 
The v-f-o frequency is given by the formula: 

freq — receiver 1st intermediate freq 
12 

r-f energy to the frequency-control circuits in the 
monitor. A reactance tube is shunted across the 
oscillator tank for fine frequency control. Oscilla- 
tor-tripler output is capacitively coupled to the 
untuned grid of the doubler tube, V803. 


WADC 7R 56-156 


310 



Section I 
Crystal Oscillators 


TO couuti TO uc fm n coaum. 

HEUTOOOt -KUDWOIO lEUtT 0602 
OOtUMOKC CHASSIS OHOUAWIIEC 



nSCIIMUTK ISOLATIW NOTOH CWTOOI. SUTPOIOT AUDIO CHASSIS H AID AUDIO 

AMPIIFIEI IELATS NODULATOfI CHASSIS 


Figure 1-198. K-f head. Block diagram of main channel transmiHer and receiver, r-f circuits in Radio Set 

AN/ARC-33 


1-492. REACTANCE TUBE. The reactance tube, 
V801, is a device that represents a variable and 
controllable reactance shunted across the tuned 
grid circuit of the variable frequency oscillator. 
Its purpose is to convert d-c control voltage from 
the monitor into a small frequency variation of 
the variable frequency oscillator. Radio-frequency 
voltages from the oscillator are coupled through 
capacitor C805 to the plate of the reactance tube 
and directly to a network made up of capacitors 
C804 and C818, resistors R801 and R804, and 
the grid-cathode capacitance of the tube, C^. 
Capacitor C804 is a blocking capacitor, so that as 
far as the a-c functioning of the tube is con- 
cerned, it need not be considered. Cathode bypass 
capacitor C803 is sufficiently large for the cathode 
to be considered at r-f ground potential. Resistor 
R802, inductor L829, and capacitors C801 and 
C802 provide a de-coupling network and filter 
through which the d-c control voltage is applied 
to the control grid without shunting its r-f input 
impedance. Resistor R807 is the plate feed com- 
ponent. Resistors R801 and R804 and capacitor 
C818 in conjunction with the grid-cathode capaci- 


tance, Cc, form a phase-shifting network such that 
the r-f voltage applied to the grid lags the oscil- 
lator voltage input to the network by 90 degrees 
approximately. Since the plate current of react- 
ance tube V801 is in phase with the grid voltage, 
the plate current flow lags the oscillator r-f volt- 
age applied to the plate by approximately 90 
degrees. This appears to the variable frequency 
oscillator as an inductive reactance since the cur- 
rent in an inductor lags the applied voltage by 
90 degrees. By controlling the d-c bias applied to 
the grid, it is possible to control the amplitude of 
the current, and thus the magnitude of the effec- 
tive inductive reactance shunting the oscillator 
tank. In this way, the frequency of the variable 
frequency oscillator can be controlled within a 
narrow range by means of a d-c control voltage. 
A positive control voltage increases the reactive 
current in the tube and thereby decreases the 
effective inductive reactance across the oscillator 
tank. The effect, therefore, is to cause an increase 
in the frequency. A negative-going control voltage 
has the opposite effect. The reactance tube is 
biased by cathode resistor R805 by an amount 


WADC TR 56-156 


311 















Section I 

Crystal Oscillators 


that permits normal “onrfrequency” oscillator 
operation when the control voltage from the mon- 
itor is approximately plus 5 volts. 

1-493. FIRST DOUBLER. The first doubler of 
the r-f head is a conventional grid-leak-biased 
frequency multiplier. The stage employs tube 
V803 with an output circuit tuned to twice the 
input frequency. The plate circuit consists of two 
inductors in parallel in order to increase their 
sizes and make it possible to employ coiled induc- 
tors rather than a more space-consuming linear 
line. The tuning is ganged with the other r-f 
tuned circuits as shown in the schematic diagram. 
Radio-frequency energy used for heterodyne oscil- 
lator injection into the main channel receiver 1st 
mixer is taken off at a tap of plate tank L802. 
The variable frequency oscillator-tripler, the re- 
actance tube, and the doubler are continuously 
supplied with plate and screen voltages since 
these circuits are employed in both transmit and 
receive operation. From the latter stage to the 
antenna, however, plate and screen voltages are 
removed from the transmitter tubes during re- 
ceive operation. 

1-494. TRANSMITTER MIXER. It is in the 
mixer that a 7.6625-mc signal from the sidestep 
oscillator is added to the first doubler output to 
make the mixer output frequency exactly one half 
of the transmitting antenna frequency. (Remem- 
ber that 7.6625 me is one half of the 1st receiver 
intermediate frequency of 15.325 me.) A conven- 
tional mixer circuit employing tube V805 is used 
here, with the side-step oscillator voltage being 
injected at the control grid through capacitor 
C821 and with the doubler output being fed to 
the control grid through capacitor C820. The 
mixer plate circuit is tuned to the sum frequency 
by means of a doubler-inductor arrangement. One 
of the inductors is wound with concentric cable, 
which allows the tuning capacitor rotor and the 
cold ends of the inductors to be returned to 
ground. The center conductor is connected to the 
plate on the hot end and is bypassed to ground 
on the cold end by capacitor C825. Plate voltage 
is supplied to tube V805 through resistor R819. 
1-495. SIDESTEP CRYSTAL OSCILLATOR. 
The sidestep crystal oscillator employs pentode 
V804, connected as a triode and operating on a 
single frequency of 7.6625 me. The circuit is 
arranged as a Pierce oscillator with the crystal 
connected directly from the plate to the grid of 
the oscillator tube. A type CR-18/U crystal unit 
is employed. Since this oscillator controls only a 
small percentage of the final transmitter fre- 
quency, its normal operating stability is sufficient 

WADC TR 56-156 


without the use of thermostatically controlled 
temperature for the crystal unit. Note that an 
18-/vif capacitor is shunted directly across the 
crystal unit and that no externally connected 
plate-to-ground capacitor is used. The 18 ^ rep- 
resents 60 per cent of the required load capaci- 
tance for the type CR-18/U crystal unit. With 
the grid-to-ground capacitance probably 8 or 10 
greater than the 68 n4 of the externally con- 
nected grid-ground capacitor C822, it appears 
that the circuit has been designed to provide a 
maximum output voltage consistent with the 
drive-level and load-capacitance requirements of 
the crystal unit, rather than for maximum fre- 
quency stability. The oscillator is fed plate voltage 
through the voltage-dropping resistor R815. The 
oscillator output is capacitively coupled from the 
plate of the oscillator tube to the control grid of 
mixer tube V805 through C821. Plate voltage is 
derived from the plus 200-volt transmitter sup- 
ply, so that the tube operates only when the radio 
set is turned to the transmit position. 

1-496. AMPLIFIER. This is a conventional grid- 
leak-biased r-f amplifier employing pentode V806. 
The circuit acts as a direct amplifier with the 
output circuit tuned to the same frequency as the 
input circuit. The input is capacitance-coupled to 
the preceding mixer stage through capacitor C828 
and the output is capacitance-coupled to doubler 
V807 through capacitor C832. The amplifier tuned 
output circuit is a double-inductor tank of the 
same type that is used in the plate circuit of the 
transmitter mixer. 

1-497. TRANSMITTER DOUBLER. The doubler, 
V807, is a grid-leak-biased stage that employs a 
cavity-tuned plate circuit tuned to twice the input 
frequency. A pencil triode tube is used because 
its small size permits superior u-h-f operating 
characteristics. 

1-498. INTERMEDIATE POWER AMPLIFIER. 
The intermediate power amplifier (IPA) employs 
a lighthouse tube, V808, in a grounded-grid cir- 
cuit with cavity tuning of the plate. The cavity is 
tuned to the doubler output frequency, providing 
an IPA output frequency equal to the transmitter 
antenna frequency. Radio-frequency input is 
coupled magnetically from the output of the pre- 
ceding doubler tube, V807, through coupling loop 
L809, which applies rf between ground and the 
IPA cathode through capacitor C838. Concentric 
inductor L810 is inserted in the heater circuit to 
make it at the same r-f potential as the cathode. 
Capacitor C838 also blo<^s the 12.6-volt d-c fila- 
ment potential from shorting to ground through 
the ground coupling loop L809. One heater con- 


312 










WADC TR 56-156 






















+400Vi AUDIO MODULATOR OUTPUT 



Figure I -199. R-f head. Scheme 
transmitter and receiver r-f ctrci 


J.-. 





















Scdton i 
Crystal Osdflators 


FlUlNfUFO 
1 m 2C39* 



Figure 1-199. K-f head. Schematic diagram of main channel 
transmitter and receiver r-f circuits in Radio Set AN/ARC-33 

c 


313-314 


nection and the cathode connection are common. 
The heater circuit is wired in series with the 
heater of the final amplifier so as to provide the 
required 12.6-volt total drop across the entire 
circuit after passing through dropping resistor 
R833. A voltage divider composed of resistors 
R830 and R832 is connected between one side of 
the filament and ground. The grid return is con- 
nected to the tap to provide the proper negative 
bias voltage. These resistors are high in value so 
that they do not influence the heater voltage and 
current. IPA output is coupled to the final power 
amplifier input through an inductive coupling 
loop, L812, in the cavity. Since the final power 
amplifier is a grounded-grid r-f amplifier, it can- 
not be plate modulated 100 per cent unless the 
output of the exciting stage also is modulated. 
Therefore, to be able to modulate fully the final 
amplifier output, the plates of doubler V807 and 
IPA V808 are modulated by the same audio volt- 
age as the power amplifier. 

1-499. FINAL POWER AMPLIFIER. The final 
r-f power amplifier, V809, is similar in design 
and operation to the intermediate power amplifier. 
Grid bias for the final amplifier is derived from 
resistor R835. Resistor R834 supplies a cathode 
bias. Radio-frequency output to the antenna is 
coupled inductively to the cavity magnetic field 
by pickup loop L815 and is fed to the antenna 
receive-transmit changeover relay through a co- 
axial cable. The coaxial connector is mounted 
directly on the cavity assembly, 0802. To provide 
a sidetone voltage that is indicative of the trans- 
mission quality, the sidetone voltage is obtained 
from the final r-f output cavity through inductive 
coupling L814. This r-f voltage is rectified through 
the sidetone crystal rectifier, CR801, and applied 
to the i-f and audio chassis to be fed through the 
receiver audio system to the operator’s headset. 
1-500. MAIN CHANNEL RECEIVER R-F CIR- 
CUITS. The main channel receiver includes two 
r-f amplifier stages, a mixer, a frequency doubler, 
and a 15.325-mc i-f amplifier. Both r-f amplifiers 
and the doubler are cavity-tuned. The doubler is 
included to multiply the output frequency of the 
first r-f-head doubler. By employing a heterodyne 
injection frequency controlled by the same vari- 
able frequency oscillator that controls the trans- 
mitter frequency, perfect tracking between the 
transmitter and receiver tuning is possible. The 
i-f output is fed by coaxial cable to the second 
mixer and the 2.8-mc intermediate frequency am- 
plifier on the i-f and audio chassis. 

1-501. FIRST R-F AMPLIFIER. The first r-f 
amplifier, V814, is a grounded-grid cavity-tuned 


Section I 
Crystal Oscillators 

amplifier with a shunt-fed plate. The cavity is 
similar to the cavities used in the transmitter sec- 
tion and operates in exactly the same way except 
that the amplifier tube is located outside of the 
cavity. Electrical connections are made through 
holes in the cavity walls. Tube V814 is tapped 
down on the cavity center column to prevent load- 
ing of the tuned circuit by the tube. The antenna 
input is also tapped down to match the character- 
istic impedance of the coaxial line. The grounded- 
grid circuit is effective in preventing coupling be- 
tween the input and output circuits, thus prevent- 
ing oscillations due to feedback through the tube 
capacitance. Also, the grounded-grid design with 
the cathode-injection input provides a more c?»n- 
stant input impedance over the frequency range. 
Two plate and two grid connections are provided 
on the JAN-6F4 tubes to make it possible to use 
shorter leads and to obtain better bypassing bal- 
ance, as is illustrated by the V814 grid connections 
to ground through capacitors C892, C893, C898, 
and C899. Examination of the tuned-cavity cir- 
cuits will show that only the first input cavity 
employs a capacitance connection, represented by 
C896. Since this cavity is not loaded by the plate 
capacitance of a vacuum tube, capacitor C896 is 
added in order for all three cavities to have sim- 
ilar tuning characteristics. Automatic-ga